Properties

Degree 48
Conductor $ 3^{24} \cdot 7^{48} \cdot 41^{24} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 24·3-s + 24·4-s − 4·5-s − 192·6-s + 24·8-s + 300·9-s − 32·10-s + 12·11-s − 576·12-s + 96·15-s − 37·16-s − 8·17-s + 2.40e3·18-s + 4·19-s − 96·20-s + 96·22-s + 20·23-s − 576·24-s − 28·25-s − 2.60e3·27-s + 24·29-s + 768·30-s + 4·31-s − 132·32-s − 288·33-s − 64·34-s + ⋯
L(s)  = 1  + 5.65·2-s − 13.8·3-s + 12·4-s − 1.78·5-s − 78.3·6-s + 8.48·8-s + 100·9-s − 10.1·10-s + 3.61·11-s − 166.·12-s + 24.7·15-s − 9.25·16-s − 1.94·17-s + 565.·18-s + 0.917·19-s − 21.4·20-s + 20.4·22-s + 4.17·23-s − 117.·24-s − 5.59·25-s − 500.·27-s + 4.45·29-s + 140.·30-s + 0.718·31-s − 23.3·32-s − 50.1·33-s − 10.9·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{48} \cdot 41^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{48} \cdot 41^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{24} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(48\)
\( N \)  =  \(3^{24} \cdot 7^{48} \cdot 41^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6027} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(48,\ 3^{24} \cdot 7^{48} \cdot 41^{24} ,\ ( \ : [1/2]^{24} ),\ 1 )$
$L(1)$  $\approx$  $129.7195367$
$L(\frac12)$  $\approx$  $129.7195367$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \(F_p\) is a polynomial of degree 48. If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 47.
$p$$F_p$
bad3 \( ( 1 + T )^{24} \)
7 \( 1 \)
41 \( ( 1 - T )^{24} \)
good2 \( 1 - p^{3} T + 5 p^{3} T^{2} - 19 p^{3} T^{3} + 485 T^{4} - 339 p^{2} T^{5} + 857 p^{2} T^{6} - 499 p^{4} T^{7} + 4347 p^{2} T^{8} - 8939 p^{2} T^{9} + 17499 p^{2} T^{10} - 32803 p^{2} T^{11} + 29591 p^{3} T^{12} - 103163 p^{2} T^{13} + 348747 p T^{14} - 71661 p^{4} T^{15} + 1838339 T^{16} - 720519 p^{2} T^{17} + 2214727 p T^{18} - 1672113 p^{2} T^{19} + 9944815 T^{20} - 911725 p^{4} T^{21} + 660857 p^{5} T^{22} - 1896015 p^{4} T^{23} + 5387865 p^{3} T^{24} - 1896015 p^{5} T^{25} + 660857 p^{7} T^{26} - 911725 p^{7} T^{27} + 9944815 p^{4} T^{28} - 1672113 p^{7} T^{29} + 2214727 p^{7} T^{30} - 720519 p^{9} T^{31} + 1838339 p^{8} T^{32} - 71661 p^{13} T^{33} + 348747 p^{11} T^{34} - 103163 p^{13} T^{35} + 29591 p^{15} T^{36} - 32803 p^{15} T^{37} + 17499 p^{16} T^{38} - 8939 p^{17} T^{39} + 4347 p^{18} T^{40} - 499 p^{21} T^{41} + 857 p^{20} T^{42} - 339 p^{21} T^{43} + 485 p^{20} T^{44} - 19 p^{24} T^{45} + 5 p^{25} T^{46} - p^{26} T^{47} + p^{24} T^{48} \)
5 \( 1 + 4 T + 44 T^{2} + 152 T^{3} + 986 T^{4} + 3004 T^{5} + 15092 T^{6} + 41512 T^{7} + 179006 T^{8} + 453812 T^{9} + 1767752 T^{10} + 4194316 T^{11} + 15183836 T^{12} + 34069548 T^{13} + 116507924 T^{14} + 49795404 p T^{15} + 812411122 T^{16} + 332436384 p T^{17} + 5209146712 T^{18} + 2049273164 p T^{19} + 6192915174 p T^{20} + 58731704856 T^{21} + 171490969868 T^{22} + 314197077572 T^{23} + 886998933014 T^{24} + 314197077572 p T^{25} + 171490969868 p^{2} T^{26} + 58731704856 p^{3} T^{27} + 6192915174 p^{5} T^{28} + 2049273164 p^{6} T^{29} + 5209146712 p^{6} T^{30} + 332436384 p^{8} T^{31} + 812411122 p^{8} T^{32} + 49795404 p^{10} T^{33} + 116507924 p^{10} T^{34} + 34069548 p^{11} T^{35} + 15183836 p^{12} T^{36} + 4194316 p^{13} T^{37} + 1767752 p^{14} T^{38} + 453812 p^{15} T^{39} + 179006 p^{16} T^{40} + 41512 p^{17} T^{41} + 15092 p^{18} T^{42} + 3004 p^{19} T^{43} + 986 p^{20} T^{44} + 152 p^{21} T^{45} + 44 p^{22} T^{46} + 4 p^{23} T^{47} + p^{24} T^{48} \)
11 \( 1 - 12 T + 160 T^{2} - 1200 T^{3} + 9853 T^{4} - 5244 p T^{5} + 377244 T^{6} - 172268 p T^{7} + 10770960 T^{8} - 48163204 T^{9} + 247364784 T^{10} - 1005614028 T^{11} + 4792363361 T^{12} - 17979181920 T^{13} + 80913385796 T^{14} - 282885004540 T^{15} + 1218530308439 T^{16} - 4007342019520 T^{17} + 16707757859916 T^{18} - 52151227456720 T^{19} + 212054076070306 T^{20} - 634072552354184 T^{21} + 2525795230246884 T^{22} - 7305640516984240 T^{23} + 28503202940977040 T^{24} - 7305640516984240 p T^{25} + 2525795230246884 p^{2} T^{26} - 634072552354184 p^{3} T^{27} + 212054076070306 p^{4} T^{28} - 52151227456720 p^{5} T^{29} + 16707757859916 p^{6} T^{30} - 4007342019520 p^{7} T^{31} + 1218530308439 p^{8} T^{32} - 282885004540 p^{9} T^{33} + 80913385796 p^{10} T^{34} - 17979181920 p^{11} T^{35} + 4792363361 p^{12} T^{36} - 1005614028 p^{13} T^{37} + 247364784 p^{14} T^{38} - 48163204 p^{15} T^{39} + 10770960 p^{16} T^{40} - 172268 p^{18} T^{41} + 377244 p^{18} T^{42} - 5244 p^{20} T^{43} + 9853 p^{20} T^{44} - 1200 p^{21} T^{45} + 160 p^{22} T^{46} - 12 p^{23} T^{47} + p^{24} T^{48} \)
13 \( 1 + 12 p T^{2} + 88 T^{3} + 12200 T^{4} + 14116 T^{5} + 640646 T^{6} + 85380 p T^{7} + 25575326 T^{8} + 57323260 T^{9} + 832146132 T^{10} + 2196586620 T^{11} + 23015839922 T^{12} + 66852705448 T^{13} + 554759184420 T^{14} + 1687447000484 T^{15} + 11816402669728 T^{16} + 36358024943716 T^{17} + 17241865917156 p T^{18} + 681538754875808 T^{19} + 3804256503081218 T^{20} + 11247446402015216 T^{21} + 57955933951247218 T^{22} + 164504136242813800 T^{23} + 794142160718931434 T^{24} + 164504136242813800 p T^{25} + 57955933951247218 p^{2} T^{26} + 11247446402015216 p^{3} T^{27} + 3804256503081218 p^{4} T^{28} + 681538754875808 p^{5} T^{29} + 17241865917156 p^{7} T^{30} + 36358024943716 p^{7} T^{31} + 11816402669728 p^{8} T^{32} + 1687447000484 p^{9} T^{33} + 554759184420 p^{10} T^{34} + 66852705448 p^{11} T^{35} + 23015839922 p^{12} T^{36} + 2196586620 p^{13} T^{37} + 832146132 p^{14} T^{38} + 57323260 p^{15} T^{39} + 25575326 p^{16} T^{40} + 85380 p^{18} T^{41} + 640646 p^{18} T^{42} + 14116 p^{19} T^{43} + 12200 p^{20} T^{44} + 88 p^{21} T^{45} + 12 p^{23} T^{46} + p^{24} T^{48} \)
17 \( 1 + 8 T + 188 T^{2} + 1304 T^{3} + 17795 T^{4} + 109632 T^{5} + 1134978 T^{6} + 6333060 T^{7} + 55008678 T^{8} + 281910156 T^{9} + 2162000470 T^{10} + 10278217684 T^{11} + 71744213731 T^{12} + 318795566476 T^{13} + 2066351457388 T^{14} + 8635299530664 T^{15} + 52707283406651 T^{16} + 208281063186496 T^{17} + 1209096959991100 T^{18} + 4540807903409428 T^{19} + 25240362595506414 T^{20} + 90510219857339396 T^{21} + 483626983180044084 T^{22} + 1662662798840358384 T^{23} + 8551041408580808820 T^{24} + 1662662798840358384 p T^{25} + 483626983180044084 p^{2} T^{26} + 90510219857339396 p^{3} T^{27} + 25240362595506414 p^{4} T^{28} + 4540807903409428 p^{5} T^{29} + 1209096959991100 p^{6} T^{30} + 208281063186496 p^{7} T^{31} + 52707283406651 p^{8} T^{32} + 8635299530664 p^{9} T^{33} + 2066351457388 p^{10} T^{34} + 318795566476 p^{11} T^{35} + 71744213731 p^{12} T^{36} + 10278217684 p^{13} T^{37} + 2162000470 p^{14} T^{38} + 281910156 p^{15} T^{39} + 55008678 p^{16} T^{40} + 6333060 p^{17} T^{41} + 1134978 p^{18} T^{42} + 109632 p^{19} T^{43} + 17795 p^{20} T^{44} + 1304 p^{21} T^{45} + 188 p^{22} T^{46} + 8 p^{23} T^{47} + p^{24} T^{48} \)
19 \( 1 - 4 T + 198 T^{2} - 716 T^{3} + 19334 T^{4} - 63224 T^{5} + 1239984 T^{6} - 3645768 T^{7} + 58726278 T^{8} - 153294568 T^{9} + 2189802808 T^{10} - 4957032832 T^{11} + 66923845690 T^{12} - 125792084348 T^{13} + 1723061310918 T^{14} - 2466679550556 T^{15} + 38182710905147 T^{16} - 33899106199592 T^{17} + 744784720887156 T^{18} - 167802365829952 T^{19} + 13239806062873424 T^{20} + 6833316909042200 T^{21} + 228056233279724584 T^{22} + 254487587325462336 T^{23} + 4132818041334698332 T^{24} + 254487587325462336 p T^{25} + 228056233279724584 p^{2} T^{26} + 6833316909042200 p^{3} T^{27} + 13239806062873424 p^{4} T^{28} - 167802365829952 p^{5} T^{29} + 744784720887156 p^{6} T^{30} - 33899106199592 p^{7} T^{31} + 38182710905147 p^{8} T^{32} - 2466679550556 p^{9} T^{33} + 1723061310918 p^{10} T^{34} - 125792084348 p^{11} T^{35} + 66923845690 p^{12} T^{36} - 4957032832 p^{13} T^{37} + 2189802808 p^{14} T^{38} - 153294568 p^{15} T^{39} + 58726278 p^{16} T^{40} - 3645768 p^{17} T^{41} + 1239984 p^{18} T^{42} - 63224 p^{19} T^{43} + 19334 p^{20} T^{44} - 716 p^{21} T^{45} + 198 p^{22} T^{46} - 4 p^{23} T^{47} + p^{24} T^{48} \)
23 \( 1 - 20 T + 468 T^{2} - 6376 T^{3} + 89492 T^{4} - 947060 T^{5} + 10031056 T^{6} - 87745472 T^{7} + 763067906 T^{8} - 5716802292 T^{9} + 42649290636 T^{10} - 280193064944 T^{11} + 1846135975040 T^{12} - 10841556025688 T^{13} + 64693138763862 T^{14} - 346565197733416 T^{15} + 1924875964270770 T^{16} - 9646116878014636 T^{17} + 51405165004399418 T^{18} - 247688418691072976 T^{19} + 1297515293839123984 T^{20} - 6121567858354523612 T^{21} + 31714018207707761568 T^{22} - \)\(14\!\cdots\!00\)\( T^{23} + \)\(74\!\cdots\!26\)\( T^{24} - \)\(14\!\cdots\!00\)\( p T^{25} + 31714018207707761568 p^{2} T^{26} - 6121567858354523612 p^{3} T^{27} + 1297515293839123984 p^{4} T^{28} - 247688418691072976 p^{5} T^{29} + 51405165004399418 p^{6} T^{30} - 9646116878014636 p^{7} T^{31} + 1924875964270770 p^{8} T^{32} - 346565197733416 p^{9} T^{33} + 64693138763862 p^{10} T^{34} - 10841556025688 p^{11} T^{35} + 1846135975040 p^{12} T^{36} - 280193064944 p^{13} T^{37} + 42649290636 p^{14} T^{38} - 5716802292 p^{15} T^{39} + 763067906 p^{16} T^{40} - 87745472 p^{17} T^{41} + 10031056 p^{18} T^{42} - 947060 p^{19} T^{43} + 89492 p^{20} T^{44} - 6376 p^{21} T^{45} + 468 p^{22} T^{46} - 20 p^{23} T^{47} + p^{24} T^{48} \)
29 \( 1 - 24 T + 606 T^{2} - 9808 T^{3} + 152695 T^{4} - 1939040 T^{5} + 23413576 T^{6} - 250734720 T^{7} + 2557413156 T^{8} - 24059297280 T^{9} + 216462578354 T^{10} - 1834720775772 T^{11} + 14934301537285 T^{12} - 115948863503400 T^{13} + 867740305010632 T^{14} - 6241449324024848 T^{15} + 43409501966193826 T^{16} - 291565021041055804 T^{17} + 1898377767526902348 T^{18} - 11972558199248363004 T^{19} + 73334509417084202527 T^{20} - 15030745014756467936 p T^{21} + \)\(25\!\cdots\!52\)\( T^{22} - \)\(14\!\cdots\!88\)\( T^{23} + \)\(77\!\cdots\!96\)\( T^{24} - \)\(14\!\cdots\!88\)\( p T^{25} + \)\(25\!\cdots\!52\)\( p^{2} T^{26} - 15030745014756467936 p^{4} T^{27} + 73334509417084202527 p^{4} T^{28} - 11972558199248363004 p^{5} T^{29} + 1898377767526902348 p^{6} T^{30} - 291565021041055804 p^{7} T^{31} + 43409501966193826 p^{8} T^{32} - 6241449324024848 p^{9} T^{33} + 867740305010632 p^{10} T^{34} - 115948863503400 p^{11} T^{35} + 14934301537285 p^{12} T^{36} - 1834720775772 p^{13} T^{37} + 216462578354 p^{14} T^{38} - 24059297280 p^{15} T^{39} + 2557413156 p^{16} T^{40} - 250734720 p^{17} T^{41} + 23413576 p^{18} T^{42} - 1939040 p^{19} T^{43} + 152695 p^{20} T^{44} - 9808 p^{21} T^{45} + 606 p^{22} T^{46} - 24 p^{23} T^{47} + p^{24} T^{48} \)
31 \( 1 - 4 T + 424 T^{2} - 1832 T^{3} + 88929 T^{4} - 401220 T^{5} + 12290986 T^{6} - 56490716 T^{7} + 1256676064 T^{8} - 5783687412 T^{9} + 101162615860 T^{10} - 14869720708 p T^{11} + 6669678121897 T^{12} - 29879959145544 T^{13} + 370495872440910 T^{14} - 1625815774752516 T^{15} + 17743348589637395 T^{16} - 2456771209972272 p T^{17} + 747985830680632162 T^{18} - 3139687339864131648 T^{19} + 28315814186115179194 T^{20} - \)\(11\!\cdots\!24\)\( T^{21} + \)\(98\!\cdots\!74\)\( T^{22} - \)\(39\!\cdots\!24\)\( T^{23} + \)\(31\!\cdots\!24\)\( T^{24} - \)\(39\!\cdots\!24\)\( p T^{25} + \)\(98\!\cdots\!74\)\( p^{2} T^{26} - \)\(11\!\cdots\!24\)\( p^{3} T^{27} + 28315814186115179194 p^{4} T^{28} - 3139687339864131648 p^{5} T^{29} + 747985830680632162 p^{6} T^{30} - 2456771209972272 p^{8} T^{31} + 17743348589637395 p^{8} T^{32} - 1625815774752516 p^{9} T^{33} + 370495872440910 p^{10} T^{34} - 29879959145544 p^{11} T^{35} + 6669678121897 p^{12} T^{36} - 14869720708 p^{14} T^{37} + 101162615860 p^{14} T^{38} - 5783687412 p^{15} T^{39} + 1256676064 p^{16} T^{40} - 56490716 p^{17} T^{41} + 12290986 p^{18} T^{42} - 401220 p^{19} T^{43} + 88929 p^{20} T^{44} - 1832 p^{21} T^{45} + 424 p^{22} T^{46} - 4 p^{23} T^{47} + p^{24} T^{48} \)
37 \( 1 - 64 T + 2410 T^{2} - 66184 T^{3} + 1464801 T^{4} - 27468288 T^{5} + 451033882 T^{6} - 6628576244 T^{7} + 88619890982 T^{8} - 1090992950820 T^{9} + 12486967959870 T^{10} - 133883024418284 T^{11} + 1353009156376243 T^{12} - 12952449366195580 T^{13} + 117941891676539554 T^{14} - 1024989901618775348 T^{15} + 8525614435667863004 T^{16} - 68028494063812806084 T^{17} + \)\(52\!\cdots\!38\)\( T^{18} - \)\(38\!\cdots\!72\)\( T^{19} + \)\(27\!\cdots\!01\)\( T^{20} - \)\(18\!\cdots\!16\)\( T^{21} + \)\(12\!\cdots\!58\)\( T^{22} - \)\(79\!\cdots\!52\)\( T^{23} + \)\(49\!\cdots\!84\)\( T^{24} - \)\(79\!\cdots\!52\)\( p T^{25} + \)\(12\!\cdots\!58\)\( p^{2} T^{26} - \)\(18\!\cdots\!16\)\( p^{3} T^{27} + \)\(27\!\cdots\!01\)\( p^{4} T^{28} - \)\(38\!\cdots\!72\)\( p^{5} T^{29} + \)\(52\!\cdots\!38\)\( p^{6} T^{30} - 68028494063812806084 p^{7} T^{31} + 8525614435667863004 p^{8} T^{32} - 1024989901618775348 p^{9} T^{33} + 117941891676539554 p^{10} T^{34} - 12952449366195580 p^{11} T^{35} + 1353009156376243 p^{12} T^{36} - 133883024418284 p^{13} T^{37} + 12486967959870 p^{14} T^{38} - 1090992950820 p^{15} T^{39} + 88619890982 p^{16} T^{40} - 6628576244 p^{17} T^{41} + 451033882 p^{18} T^{42} - 27468288 p^{19} T^{43} + 1464801 p^{20} T^{44} - 66184 p^{21} T^{45} + 2410 p^{22} T^{46} - 64 p^{23} T^{47} + p^{24} T^{48} \)
43 \( 1 - 20 T + 710 T^{2} - 11176 T^{3} + 235105 T^{4} - 3127316 T^{5} + 1160750 p T^{6} - 582695556 T^{7} + 7734792124 T^{8} - 81134917976 T^{9} + 938682131306 T^{10} - 8989775513148 T^{11} + 93227588511509 T^{12} - 824691023564864 T^{13} + 7810024975642798 T^{14} - 64379119306803456 T^{15} + 564153470822562895 T^{16} - 4363205069082300540 T^{17} + 35723254736393062000 T^{18} - \)\(26\!\cdots\!84\)\( T^{19} + \)\(20\!\cdots\!78\)\( T^{20} - \)\(13\!\cdots\!00\)\( T^{21} + \)\(10\!\cdots\!04\)\( T^{22} - \)\(66\!\cdots\!64\)\( T^{23} + \)\(45\!\cdots\!84\)\( T^{24} - \)\(66\!\cdots\!64\)\( p T^{25} + \)\(10\!\cdots\!04\)\( p^{2} T^{26} - \)\(13\!\cdots\!00\)\( p^{3} T^{27} + \)\(20\!\cdots\!78\)\( p^{4} T^{28} - \)\(26\!\cdots\!84\)\( p^{5} T^{29} + 35723254736393062000 p^{6} T^{30} - 4363205069082300540 p^{7} T^{31} + 564153470822562895 p^{8} T^{32} - 64379119306803456 p^{9} T^{33} + 7810024975642798 p^{10} T^{34} - 824691023564864 p^{11} T^{35} + 93227588511509 p^{12} T^{36} - 8989775513148 p^{13} T^{37} + 938682131306 p^{14} T^{38} - 81134917976 p^{15} T^{39} + 7734792124 p^{16} T^{40} - 582695556 p^{17} T^{41} + 1160750 p^{19} T^{42} - 3127316 p^{19} T^{43} + 235105 p^{20} T^{44} - 11176 p^{21} T^{45} + 710 p^{22} T^{46} - 20 p^{23} T^{47} + p^{24} T^{48} \)
47 \( 1 + 32 T + 920 T^{2} + 18576 T^{3} + 341593 T^{4} + 5345464 T^{5} + 77816162 T^{6} + 1024803084 T^{7} + 12738919608 T^{8} + 3133948956 p T^{9} + 1623560179420 T^{10} + 16906207970396 T^{11} + 169019733658681 T^{12} + 1612185449248700 T^{13} + 14844923746401944 T^{14} + 131322844534076476 T^{15} + 1126473603791951370 T^{16} + 9332206254742145228 T^{17} + 75247733615143909586 T^{18} + \)\(58\!\cdots\!68\)\( T^{19} + \)\(44\!\cdots\!65\)\( T^{20} + \)\(33\!\cdots\!24\)\( T^{21} + \)\(24\!\cdots\!12\)\( T^{22} + \)\(17\!\cdots\!40\)\( T^{23} + \)\(11\!\cdots\!36\)\( T^{24} + \)\(17\!\cdots\!40\)\( p T^{25} + \)\(24\!\cdots\!12\)\( p^{2} T^{26} + \)\(33\!\cdots\!24\)\( p^{3} T^{27} + \)\(44\!\cdots\!65\)\( p^{4} T^{28} + \)\(58\!\cdots\!68\)\( p^{5} T^{29} + 75247733615143909586 p^{6} T^{30} + 9332206254742145228 p^{7} T^{31} + 1126473603791951370 p^{8} T^{32} + 131322844534076476 p^{9} T^{33} + 14844923746401944 p^{10} T^{34} + 1612185449248700 p^{11} T^{35} + 169019733658681 p^{12} T^{36} + 16906207970396 p^{13} T^{37} + 1623560179420 p^{14} T^{38} + 3133948956 p^{16} T^{39} + 12738919608 p^{16} T^{40} + 1024803084 p^{17} T^{41} + 77816162 p^{18} T^{42} + 5345464 p^{19} T^{43} + 341593 p^{20} T^{44} + 18576 p^{21} T^{45} + 920 p^{22} T^{46} + 32 p^{23} T^{47} + p^{24} T^{48} \)
53 \( 1 - 76 T + 3478 T^{2} - 117148 T^{3} + 3204508 T^{4} - 74637244 T^{5} + 1527552248 T^{6} - 28037909400 T^{7} + 468585060464 T^{8} - 7210516298892 T^{9} + 103061709293264 T^{10} - 1377783278324036 T^{11} + 17324783250437502 T^{12} - 205852158553422904 T^{13} + 2320110449965456824 T^{14} - 24883405445207312408 T^{15} + \)\(25\!\cdots\!18\)\( T^{16} - \)\(24\!\cdots\!88\)\( T^{17} + \)\(23\!\cdots\!06\)\( T^{18} - \)\(21\!\cdots\!04\)\( T^{19} + \)\(18\!\cdots\!90\)\( T^{20} - \)\(15\!\cdots\!20\)\( T^{21} + \)\(12\!\cdots\!20\)\( T^{22} - \)\(92\!\cdots\!68\)\( T^{23} + \)\(68\!\cdots\!74\)\( T^{24} - \)\(92\!\cdots\!68\)\( p T^{25} + \)\(12\!\cdots\!20\)\( p^{2} T^{26} - \)\(15\!\cdots\!20\)\( p^{3} T^{27} + \)\(18\!\cdots\!90\)\( p^{4} T^{28} - \)\(21\!\cdots\!04\)\( p^{5} T^{29} + \)\(23\!\cdots\!06\)\( p^{6} T^{30} - \)\(24\!\cdots\!88\)\( p^{7} T^{31} + \)\(25\!\cdots\!18\)\( p^{8} T^{32} - 24883405445207312408 p^{9} T^{33} + 2320110449965456824 p^{10} T^{34} - 205852158553422904 p^{11} T^{35} + 17324783250437502 p^{12} T^{36} - 1377783278324036 p^{13} T^{37} + 103061709293264 p^{14} T^{38} - 7210516298892 p^{15} T^{39} + 468585060464 p^{16} T^{40} - 28037909400 p^{17} T^{41} + 1527552248 p^{18} T^{42} - 74637244 p^{19} T^{43} + 3204508 p^{20} T^{44} - 117148 p^{21} T^{45} + 3478 p^{22} T^{46} - 76 p^{23} T^{47} + p^{24} T^{48} \)
59 \( 1 + 28 T + 926 T^{2} + 18896 T^{3} + 393708 T^{4} + 6566908 T^{5} + 108038202 T^{6} + 1557824188 T^{7} + 21931111270 T^{8} + 282264880820 T^{9} + 3537538745054 T^{10} + 41468455521108 T^{11} + 473345711718192 T^{12} + 5122620471126304 T^{13} + 54029849016052062 T^{14} + 544900929384758524 T^{15} + 5361688582570829147 T^{16} + 50723851985941297376 T^{17} + \)\(46\!\cdots\!00\)\( T^{18} + \)\(41\!\cdots\!68\)\( T^{19} + \)\(36\!\cdots\!52\)\( T^{20} + \)\(30\!\cdots\!68\)\( T^{21} + \)\(25\!\cdots\!92\)\( T^{22} + \)\(20\!\cdots\!12\)\( T^{23} + \)\(15\!\cdots\!96\)\( T^{24} + \)\(20\!\cdots\!12\)\( p T^{25} + \)\(25\!\cdots\!92\)\( p^{2} T^{26} + \)\(30\!\cdots\!68\)\( p^{3} T^{27} + \)\(36\!\cdots\!52\)\( p^{4} T^{28} + \)\(41\!\cdots\!68\)\( p^{5} T^{29} + \)\(46\!\cdots\!00\)\( p^{6} T^{30} + 50723851985941297376 p^{7} T^{31} + 5361688582570829147 p^{8} T^{32} + 544900929384758524 p^{9} T^{33} + 54029849016052062 p^{10} T^{34} + 5122620471126304 p^{11} T^{35} + 473345711718192 p^{12} T^{36} + 41468455521108 p^{13} T^{37} + 3537538745054 p^{14} T^{38} + 282264880820 p^{15} T^{39} + 21931111270 p^{16} T^{40} + 1557824188 p^{17} T^{41} + 108038202 p^{18} T^{42} + 6566908 p^{19} T^{43} + 393708 p^{20} T^{44} + 18896 p^{21} T^{45} + 926 p^{22} T^{46} + 28 p^{23} T^{47} + p^{24} T^{48} \)
61 \( 1 - 28 T + 1014 T^{2} - 21296 T^{3} + 471709 T^{4} - 8101916 T^{5} + 139221430 T^{6} - 2055231340 T^{7} + 29865429024 T^{8} - 391093620908 T^{9} + 5017874436044 T^{10} - 59550466914108 T^{11} + 692013601854741 T^{12} - 7556214769209080 T^{13} + 80855582137451816 T^{14} - 821269437090000604 T^{15} + 8184729596182026767 T^{16} - 77955976342330886872 T^{17} + \)\(72\!\cdots\!86\)\( T^{18} - \)\(65\!\cdots\!00\)\( T^{19} + \)\(57\!\cdots\!34\)\( T^{20} - \)\(49\!\cdots\!96\)\( T^{21} + \)\(41\!\cdots\!22\)\( T^{22} - \)\(33\!\cdots\!20\)\( T^{23} + \)\(26\!\cdots\!96\)\( T^{24} - \)\(33\!\cdots\!20\)\( p T^{25} + \)\(41\!\cdots\!22\)\( p^{2} T^{26} - \)\(49\!\cdots\!96\)\( p^{3} T^{27} + \)\(57\!\cdots\!34\)\( p^{4} T^{28} - \)\(65\!\cdots\!00\)\( p^{5} T^{29} + \)\(72\!\cdots\!86\)\( p^{6} T^{30} - 77955976342330886872 p^{7} T^{31} + 8184729596182026767 p^{8} T^{32} - 821269437090000604 p^{9} T^{33} + 80855582137451816 p^{10} T^{34} - 7556214769209080 p^{11} T^{35} + 692013601854741 p^{12} T^{36} - 59550466914108 p^{13} T^{37} + 5017874436044 p^{14} T^{38} - 391093620908 p^{15} T^{39} + 29865429024 p^{16} T^{40} - 2055231340 p^{17} T^{41} + 139221430 p^{18} T^{42} - 8101916 p^{19} T^{43} + 471709 p^{20} T^{44} - 21296 p^{21} T^{45} + 1014 p^{22} T^{46} - 28 p^{23} T^{47} + p^{24} T^{48} \)
67 \( 1 - 44 T + 2082 T^{2} - 61420 T^{3} + 1760794 T^{4} - 40144116 T^{5} + 874431046 T^{6} - 16482212284 T^{7} + 296438466596 T^{8} - 71676762560 p T^{9} + 74419171365340 T^{10} - 1062442739216192 T^{11} + 14557618914427800 T^{12} - 186385350534346440 T^{13} + 2298034853240831964 T^{14} - 26730894550898825592 T^{15} + \)\(30\!\cdots\!04\)\( T^{16} - \)\(32\!\cdots\!00\)\( T^{17} + \)\(33\!\cdots\!42\)\( T^{18} - \)\(32\!\cdots\!92\)\( T^{19} + \)\(31\!\cdots\!34\)\( T^{20} - \)\(28\!\cdots\!76\)\( T^{21} + \)\(25\!\cdots\!54\)\( T^{22} - \)\(22\!\cdots\!20\)\( T^{23} + \)\(18\!\cdots\!82\)\( T^{24} - \)\(22\!\cdots\!20\)\( p T^{25} + \)\(25\!\cdots\!54\)\( p^{2} T^{26} - \)\(28\!\cdots\!76\)\( p^{3} T^{27} + \)\(31\!\cdots\!34\)\( p^{4} T^{28} - \)\(32\!\cdots\!92\)\( p^{5} T^{29} + \)\(33\!\cdots\!42\)\( p^{6} T^{30} - \)\(32\!\cdots\!00\)\( p^{7} T^{31} + \)\(30\!\cdots\!04\)\( p^{8} T^{32} - 26730894550898825592 p^{9} T^{33} + 2298034853240831964 p^{10} T^{34} - 186385350534346440 p^{11} T^{35} + 14557618914427800 p^{12} T^{36} - 1062442739216192 p^{13} T^{37} + 74419171365340 p^{14} T^{38} - 71676762560 p^{16} T^{39} + 296438466596 p^{16} T^{40} - 16482212284 p^{17} T^{41} + 874431046 p^{18} T^{42} - 40144116 p^{19} T^{43} + 1760794 p^{20} T^{44} - 61420 p^{21} T^{45} + 2082 p^{22} T^{46} - 44 p^{23} T^{47} + p^{24} T^{48} \)
71 \( 1 - 20 T + 1036 T^{2} - 17960 T^{3} + 520679 T^{4} - 7941520 T^{5} + 168066730 T^{6} - 2280945128 T^{7} + 38908864158 T^{8} - 473275872492 T^{9} + 96150520516 p T^{10} - 74637566488708 T^{11} + 933148762031291 T^{12} - 9147625711492636 T^{13} + 100176552202380218 T^{14} - 870955708304401696 T^{15} + 8332972264174495643 T^{16} - 62423915287421079324 T^{17} + \)\(50\!\cdots\!06\)\( T^{18} - \)\(30\!\cdots\!16\)\( T^{19} + \)\(19\!\cdots\!42\)\( T^{20} - \)\(53\!\cdots\!04\)\( T^{21} + \)\(16\!\cdots\!10\)\( T^{22} + \)\(48\!\cdots\!16\)\( T^{23} - \)\(40\!\cdots\!28\)\( T^{24} + \)\(48\!\cdots\!16\)\( p T^{25} + \)\(16\!\cdots\!10\)\( p^{2} T^{26} - \)\(53\!\cdots\!04\)\( p^{3} T^{27} + \)\(19\!\cdots\!42\)\( p^{4} T^{28} - \)\(30\!\cdots\!16\)\( p^{5} T^{29} + \)\(50\!\cdots\!06\)\( p^{6} T^{30} - 62423915287421079324 p^{7} T^{31} + 8332972264174495643 p^{8} T^{32} - 870955708304401696 p^{9} T^{33} + 100176552202380218 p^{10} T^{34} - 9147625711492636 p^{11} T^{35} + 933148762031291 p^{12} T^{36} - 74637566488708 p^{13} T^{37} + 96150520516 p^{15} T^{38} - 473275872492 p^{15} T^{39} + 38908864158 p^{16} T^{40} - 2280945128 p^{17} T^{41} + 168066730 p^{18} T^{42} - 7941520 p^{19} T^{43} + 520679 p^{20} T^{44} - 17960 p^{21} T^{45} + 1036 p^{22} T^{46} - 20 p^{23} T^{47} + p^{24} T^{48} \)
73 \( 1 - 16 T + 1250 T^{2} - 18388 T^{3} + 763125 T^{4} - 10425476 T^{5} + 303729258 T^{6} - 3883773420 T^{7} + 88703929704 T^{8} - 1067959721236 T^{9} + 20274650959004 T^{10} - 230825006864868 T^{11} + 3775536599664281 T^{12} - 40762780658108268 T^{13} + 588517888369686488 T^{14} - 6034621792916491064 T^{15} + 78256699073843703379 T^{16} - \)\(76\!\cdots\!16\)\( T^{17} + \)\(89\!\cdots\!46\)\( T^{18} - \)\(83\!\cdots\!96\)\( T^{19} + \)\(90\!\cdots\!26\)\( T^{20} - \)\(79\!\cdots\!68\)\( T^{21} + \)\(79\!\cdots\!26\)\( T^{22} - \)\(65\!\cdots\!20\)\( T^{23} + \)\(61\!\cdots\!20\)\( T^{24} - \)\(65\!\cdots\!20\)\( p T^{25} + \)\(79\!\cdots\!26\)\( p^{2} T^{26} - \)\(79\!\cdots\!68\)\( p^{3} T^{27} + \)\(90\!\cdots\!26\)\( p^{4} T^{28} - \)\(83\!\cdots\!96\)\( p^{5} T^{29} + \)\(89\!\cdots\!46\)\( p^{6} T^{30} - \)\(76\!\cdots\!16\)\( p^{7} T^{31} + 78256699073843703379 p^{8} T^{32} - 6034621792916491064 p^{9} T^{33} + 588517888369686488 p^{10} T^{34} - 40762780658108268 p^{11} T^{35} + 3775536599664281 p^{12} T^{36} - 230825006864868 p^{13} T^{37} + 20274650959004 p^{14} T^{38} - 1067959721236 p^{15} T^{39} + 88703929704 p^{16} T^{40} - 3883773420 p^{17} T^{41} + 303729258 p^{18} T^{42} - 10425476 p^{19} T^{43} + 763125 p^{20} T^{44} - 18388 p^{21} T^{45} + 1250 p^{22} T^{46} - 16 p^{23} T^{47} + p^{24} T^{48} \)
79 \( 1 - 4 T + 924 T^{2} - 3248 T^{3} + 445750 T^{4} - 1437920 T^{5} + 148054866 T^{6} - 448311320 T^{7} + 37742341564 T^{8} - 108673661740 T^{9} + 7820830313674 T^{10} - 21568775624376 T^{11} + 1364068419923332 T^{12} - 3615601258366608 T^{13} + 204890149909251506 T^{14} - 522444321360380612 T^{15} + 26923280164115453016 T^{16} - 65988615068493091768 T^{17} + \)\(31\!\cdots\!50\)\( T^{18} - \)\(73\!\cdots\!60\)\( T^{19} + \)\(32\!\cdots\!34\)\( T^{20} - \)\(72\!\cdots\!52\)\( T^{21} + \)\(30\!\cdots\!36\)\( T^{22} - \)\(64\!\cdots\!16\)\( T^{23} + \)\(25\!\cdots\!26\)\( T^{24} - \)\(64\!\cdots\!16\)\( p T^{25} + \)\(30\!\cdots\!36\)\( p^{2} T^{26} - \)\(72\!\cdots\!52\)\( p^{3} T^{27} + \)\(32\!\cdots\!34\)\( p^{4} T^{28} - \)\(73\!\cdots\!60\)\( p^{5} T^{29} + \)\(31\!\cdots\!50\)\( p^{6} T^{30} - 65988615068493091768 p^{7} T^{31} + 26923280164115453016 p^{8} T^{32} - 522444321360380612 p^{9} T^{33} + 204890149909251506 p^{10} T^{34} - 3615601258366608 p^{11} T^{35} + 1364068419923332 p^{12} T^{36} - 21568775624376 p^{13} T^{37} + 7820830313674 p^{14} T^{38} - 108673661740 p^{15} T^{39} + 37742341564 p^{16} T^{40} - 448311320 p^{17} T^{41} + 148054866 p^{18} T^{42} - 1437920 p^{19} T^{43} + 445750 p^{20} T^{44} - 3248 p^{21} T^{45} + 924 p^{22} T^{46} - 4 p^{23} T^{47} + p^{24} T^{48} \)
83 \( 1 + 8 T + 1052 T^{2} + 8192 T^{3} + 565016 T^{4} + 4280248 T^{5} + 205379836 T^{6} + 1509577744 T^{7} + 56577042570 T^{8} + 402214296240 T^{9} + 12549044576956 T^{10} + 86022776121288 T^{11} + 2326213367719704 T^{12} + 15333710437397056 T^{13} + 369436891475152828 T^{14} + 2336180284483928696 T^{15} + 51145780737222961839 T^{16} + \)\(30\!\cdots\!32\)\( T^{17} + \)\(62\!\cdots\!40\)\( T^{18} + \)\(36\!\cdots\!36\)\( T^{19} + \)\(67\!\cdots\!72\)\( T^{20} + \)\(37\!\cdots\!88\)\( T^{21} + \)\(66\!\cdots\!68\)\( T^{22} + \)\(34\!\cdots\!32\)\( T^{23} + \)\(57\!\cdots\!52\)\( T^{24} + \)\(34\!\cdots\!32\)\( p T^{25} + \)\(66\!\cdots\!68\)\( p^{2} T^{26} + \)\(37\!\cdots\!88\)\( p^{3} T^{27} + \)\(67\!\cdots\!72\)\( p^{4} T^{28} + \)\(36\!\cdots\!36\)\( p^{5} T^{29} + \)\(62\!\cdots\!40\)\( p^{6} T^{30} + \)\(30\!\cdots\!32\)\( p^{7} T^{31} + 51145780737222961839 p^{8} T^{32} + 2336180284483928696 p^{9} T^{33} + 369436891475152828 p^{10} T^{34} + 15333710437397056 p^{11} T^{35} + 2326213367719704 p^{12} T^{36} + 86022776121288 p^{13} T^{37} + 12549044576956 p^{14} T^{38} + 402214296240 p^{15} T^{39} + 56577042570 p^{16} T^{40} + 1509577744 p^{17} T^{41} + 205379836 p^{18} T^{42} + 4280248 p^{19} T^{43} + 565016 p^{20} T^{44} + 8192 p^{21} T^{45} + 1052 p^{22} T^{46} + 8 p^{23} T^{47} + p^{24} T^{48} \)
89 \( 1 + 60 T + 2850 T^{2} + 96480 T^{3} + 2823712 T^{4} + 69716184 T^{5} + 1547935890 T^{6} + 30620255164 T^{7} + 556135917954 T^{8} + 9220340426684 T^{9} + 141902974717170 T^{10} + 2015541058129176 T^{11} + 26676310256449264 T^{12} + 326283916673072288 T^{13} + 3700482483286046610 T^{14} + 38296814028637129788 T^{15} + \)\(35\!\cdots\!27\)\( T^{16} + \)\(28\!\cdots\!16\)\( T^{17} + \)\(17\!\cdots\!92\)\( T^{18} + \)\(41\!\cdots\!28\)\( T^{19} - \)\(10\!\cdots\!12\)\( T^{20} - \)\(23\!\cdots\!36\)\( T^{21} - \)\(33\!\cdots\!92\)\( T^{22} - \)\(39\!\cdots\!32\)\( T^{23} - \)\(39\!\cdots\!48\)\( T^{24} - \)\(39\!\cdots\!32\)\( p T^{25} - \)\(33\!\cdots\!92\)\( p^{2} T^{26} - \)\(23\!\cdots\!36\)\( p^{3} T^{27} - \)\(10\!\cdots\!12\)\( p^{4} T^{28} + \)\(41\!\cdots\!28\)\( p^{5} T^{29} + \)\(17\!\cdots\!92\)\( p^{6} T^{30} + \)\(28\!\cdots\!16\)\( p^{7} T^{31} + \)\(35\!\cdots\!27\)\( p^{8} T^{32} + 38296814028637129788 p^{9} T^{33} + 3700482483286046610 p^{10} T^{34} + 326283916673072288 p^{11} T^{35} + 26676310256449264 p^{12} T^{36} + 2015541058129176 p^{13} T^{37} + 141902974717170 p^{14} T^{38} + 9220340426684 p^{15} T^{39} + 556135917954 p^{16} T^{40} + 30620255164 p^{17} T^{41} + 1547935890 p^{18} T^{42} + 69716184 p^{19} T^{43} + 2823712 p^{20} T^{44} + 96480 p^{21} T^{45} + 2850 p^{22} T^{46} + 60 p^{23} T^{47} + p^{24} T^{48} \)
97 \( 1 - 48 T + 2276 T^{2} - 70044 T^{3} + 2024738 T^{4} - 47402412 T^{5} + 1043536788 T^{6} - 20080658276 T^{7} + 366163731746 T^{8} - 6059145574672 T^{9} + 95828828269760 T^{10} - 1407181862709864 T^{11} + 19918276779718214 T^{12} - 2742744987904680 p T^{13} + 3452955709338204722 T^{14} - 42808552312157615360 T^{15} + \)\(51\!\cdots\!68\)\( T^{16} - \)\(60\!\cdots\!48\)\( T^{17} + \)\(69\!\cdots\!02\)\( T^{18} - \)\(77\!\cdots\!76\)\( T^{19} + \)\(84\!\cdots\!44\)\( T^{20} - \)\(90\!\cdots\!52\)\( T^{21} + \)\(94\!\cdots\!52\)\( T^{22} - \)\(95\!\cdots\!28\)\( T^{23} + \)\(95\!\cdots\!42\)\( T^{24} - \)\(95\!\cdots\!28\)\( p T^{25} + \)\(94\!\cdots\!52\)\( p^{2} T^{26} - \)\(90\!\cdots\!52\)\( p^{3} T^{27} + \)\(84\!\cdots\!44\)\( p^{4} T^{28} - \)\(77\!\cdots\!76\)\( p^{5} T^{29} + \)\(69\!\cdots\!02\)\( p^{6} T^{30} - \)\(60\!\cdots\!48\)\( p^{7} T^{31} + \)\(51\!\cdots\!68\)\( p^{8} T^{32} - 42808552312157615360 p^{9} T^{33} + 3452955709338204722 p^{10} T^{34} - 2742744987904680 p^{12} T^{35} + 19918276779718214 p^{12} T^{36} - 1407181862709864 p^{13} T^{37} + 95828828269760 p^{14} T^{38} - 6059145574672 p^{15} T^{39} + 366163731746 p^{16} T^{40} - 20080658276 p^{17} T^{41} + 1043536788 p^{18} T^{42} - 47402412 p^{19} T^{43} + 2024738 p^{20} T^{44} - 70044 p^{21} T^{45} + 2276 p^{22} T^{46} - 48 p^{23} T^{47} + p^{24} T^{48} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.23704346066914761973163286684, −1.19814100491402109706838340286, −1.16924115965295672259605224900, −1.04036081309867369232949880126, −1.03993478179515962516238115299, −0.970949719015297979626417013469, −0.965807436635123582488555578292, −0.936592332778388759992096938172, −0.866504916926109090617586889504, −0.846234258579901223435554344069, −0.75024264805499257648189181871, −0.71431518586463662859576292317, −0.65106801890509515454723424798, −0.59816492981528670534481558681, −0.56018605837779275550775265647, −0.54489654852909615213263963486, −0.53401906489958562517152301149, −0.50457115338464414344420919581, −0.48075617908203773384736026197, −0.45327296601441005232904729583, −0.41319858891938600148978072086, −0.40140645872527802178729847141, −0.37607161946350356870513374068, −0.27278369691376500543557643410, −0.099801108460767392393340632926, 0.099801108460767392393340632926, 0.27278369691376500543557643410, 0.37607161946350356870513374068, 0.40140645872527802178729847141, 0.41319858891938600148978072086, 0.45327296601441005232904729583, 0.48075617908203773384736026197, 0.50457115338464414344420919581, 0.53401906489958562517152301149, 0.54489654852909615213263963486, 0.56018605837779275550775265647, 0.59816492981528670534481558681, 0.65106801890509515454723424798, 0.71431518586463662859576292317, 0.75024264805499257648189181871, 0.846234258579901223435554344069, 0.866504916926109090617586889504, 0.936592332778388759992096938172, 0.965807436635123582488555578292, 0.970949719015297979626417013469, 1.03993478179515962516238115299, 1.04036081309867369232949880126, 1.16924115965295672259605224900, 1.19814100491402109706838340286, 1.23704346066914761973163286684

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.