Properties

Degree 64
Conductor $ 13^{32} \cdot 31^{32} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 14·4-s − 30·9-s − 56·12-s + 10·13-s + 87·16-s − 8·17-s − 8·23-s + 82·25-s + 152·27-s − 16·29-s − 420·36-s − 40·39-s − 32·43-s − 348·48-s + 80·49-s + 32·51-s + 140·52-s − 12·53-s + 16·61-s + 300·64-s − 112·68-s + 32·69-s − 328·75-s + 64·79-s + 367·81-s + 64·87-s + ⋯
L(s)  = 1  − 2.30·3-s + 7·4-s − 10·9-s − 16.1·12-s + 2.77·13-s + 87/4·16-s − 1.94·17-s − 1.66·23-s + 82/5·25-s + 29.2·27-s − 2.97·29-s − 70·36-s − 6.40·39-s − 4.87·43-s − 50.2·48-s + 80/7·49-s + 4.48·51-s + 19.4·52-s − 1.64·53-s + 2.04·61-s + 75/2·64-s − 13.5·68-s + 3.85·69-s − 37.8·75-s + 7.20·79-s + 40.7·81-s + 6.86·87-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(13^{32} \cdot 31^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{32} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(13^{32} \cdot 31^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{32} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(64\)
\( N \)  =  \(13^{32} \cdot 31^{32}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{403} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(64,\ 13^{32} \cdot 31^{32} ,\ ( \ : [1/2]^{32} ),\ 1 )$
$L(1)$  $\approx$  $35.3092$
$L(\frac12)$  $\approx$  $35.3092$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;31\}$, \(F_p(T)\) is a polynomial of degree 64. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 63.
$p$$F_p(T)$
bad13 \( 1 - 10 T + 54 T^{2} - 246 T^{3} + 768 T^{4} - 466 T^{5} - 46 p^{2} T^{6} + 4042 p T^{7} - 14948 p T^{8} + 230606 T^{9} + 1652646 T^{10} - 13368494 T^{11} + 56712256 T^{12} - 127557466 T^{13} - 74936926 T^{14} + 1987807018 T^{15} - 9653765658 T^{16} + 1987807018 p T^{17} - 74936926 p^{2} T^{18} - 127557466 p^{3} T^{19} + 56712256 p^{4} T^{20} - 13368494 p^{5} T^{21} + 1652646 p^{6} T^{22} + 230606 p^{7} T^{23} - 14948 p^{9} T^{24} + 4042 p^{10} T^{25} - 46 p^{12} T^{26} - 466 p^{11} T^{27} + 768 p^{12} T^{28} - 246 p^{13} T^{29} + 54 p^{14} T^{30} - 10 p^{15} T^{31} + p^{16} T^{32} \)
31 \( ( 1 + T^{2} )^{16} \)
good2 \( 1 - 7 p T^{2} + 109 T^{4} - 19 p^{5} T^{6} + 679 p^{2} T^{8} - 323 p^{5} T^{10} + 8723 p^{2} T^{12} - 26835 p^{2} T^{14} + 153333 p T^{16} - 412229 p T^{18} + 263251 p^{3} T^{20} - 1286709 p^{2} T^{22} + 94471 p^{7} T^{24} - 213973 p^{7} T^{26} + 1871039 p^{5} T^{28} - 31583763 p^{2} T^{30} + 257236377 T^{32} - 31583763 p^{4} T^{34} + 1871039 p^{9} T^{36} - 213973 p^{13} T^{38} + 94471 p^{15} T^{40} - 1286709 p^{12} T^{42} + 263251 p^{15} T^{44} - 412229 p^{15} T^{46} + 153333 p^{17} T^{48} - 26835 p^{20} T^{50} + 8723 p^{22} T^{52} - 323 p^{27} T^{54} + 679 p^{26} T^{56} - 19 p^{31} T^{58} + 109 p^{28} T^{60} - 7 p^{31} T^{62} + p^{32} T^{64} \)
3 \( ( 1 + 2 T + 7 p T^{2} + 34 T^{3} + 218 T^{4} + 100 p T^{5} + 1555 T^{6} + 628 p T^{7} + 8731 T^{8} + 9520 T^{9} + 41144 T^{10} + 41066 T^{11} + 167989 T^{12} + 155486 T^{13} + 603142 T^{14} + 522428 T^{15} + 1919026 T^{16} + 522428 p T^{17} + 603142 p^{2} T^{18} + 155486 p^{3} T^{19} + 167989 p^{4} T^{20} + 41066 p^{5} T^{21} + 41144 p^{6} T^{22} + 9520 p^{7} T^{23} + 8731 p^{8} T^{24} + 628 p^{10} T^{25} + 1555 p^{10} T^{26} + 100 p^{12} T^{27} + 218 p^{12} T^{28} + 34 p^{13} T^{29} + 7 p^{15} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
5 \( 1 - 82 T^{2} + 3379 T^{4} - 93376 T^{6} + 77893 p^{2} T^{8} - 32686024 T^{10} + 459754421 T^{12} - 5569931124 T^{14} + 59277848901 T^{16} - 562373807296 T^{18} + 4809528013434 T^{20} - 37399793106408 T^{22} + 266205370650999 T^{24} - 1743211739493106 T^{26} + 10541350848544222 T^{28} - 59016710772591184 T^{30} + 306367369352086309 T^{32} - 59016710772591184 p^{2} T^{34} + 10541350848544222 p^{4} T^{36} - 1743211739493106 p^{6} T^{38} + 266205370650999 p^{8} T^{40} - 37399793106408 p^{10} T^{42} + 4809528013434 p^{12} T^{44} - 562373807296 p^{14} T^{46} + 59277848901 p^{16} T^{48} - 5569931124 p^{18} T^{50} + 459754421 p^{20} T^{52} - 32686024 p^{22} T^{54} + 77893 p^{26} T^{56} - 93376 p^{26} T^{58} + 3379 p^{28} T^{60} - 82 p^{30} T^{62} + p^{32} T^{64} \)
7 \( 1 - 80 T^{2} + 3184 T^{4} - 83980 T^{6} + 1651368 T^{8} - 25823842 T^{10} + 334423524 T^{12} - 3687257984 T^{14} + 5050249530 p T^{16} - 300527510464 T^{18} + 2317926878285 T^{20} - 16733179172714 T^{22} + 16798338322032 p T^{24} - 833400278004274 T^{26} + 861143982948948 p T^{28} - 43773938950481500 T^{30} + 311325781434187773 T^{32} - 43773938950481500 p^{2} T^{34} + 861143982948948 p^{5} T^{36} - 833400278004274 p^{6} T^{38} + 16798338322032 p^{9} T^{40} - 16733179172714 p^{10} T^{42} + 2317926878285 p^{12} T^{44} - 300527510464 p^{14} T^{46} + 5050249530 p^{17} T^{48} - 3687257984 p^{18} T^{50} + 334423524 p^{20} T^{52} - 25823842 p^{22} T^{54} + 1651368 p^{24} T^{56} - 83980 p^{26} T^{58} + 3184 p^{28} T^{60} - 80 p^{30} T^{62} + p^{32} T^{64} \)
11 \( 1 - 190 T^{2} + 17885 T^{4} - 1112248 T^{6} + 51422662 T^{8} - 1886188066 T^{10} + 57222048046 T^{12} - 1478501326916 T^{14} + 33263831415423 T^{16} - 663205861468002 T^{18} + 1080742999399334 p T^{20} - 193902928129004992 T^{22} + 24019851646762491 p^{2} T^{24} - 40352122153928171750 T^{26} + \)\(52\!\cdots\!91\)\( T^{28} - \)\(63\!\cdots\!64\)\( T^{30} + \)\(71\!\cdots\!86\)\( T^{32} - \)\(63\!\cdots\!64\)\( p^{2} T^{34} + \)\(52\!\cdots\!91\)\( p^{4} T^{36} - 40352122153928171750 p^{6} T^{38} + 24019851646762491 p^{10} T^{40} - 193902928129004992 p^{10} T^{42} + 1080742999399334 p^{13} T^{44} - 663205861468002 p^{14} T^{46} + 33263831415423 p^{16} T^{48} - 1478501326916 p^{18} T^{50} + 57222048046 p^{20} T^{52} - 1886188066 p^{22} T^{54} + 51422662 p^{24} T^{56} - 1112248 p^{26} T^{58} + 17885 p^{28} T^{60} - 190 p^{30} T^{62} + p^{32} T^{64} \)
17 \( ( 1 + 4 T + 167 T^{2} + 638 T^{3} + 14230 T^{4} + 51436 T^{5} + 811354 T^{6} + 2759804 T^{7} + 34419993 T^{8} + 109711108 T^{9} + 1147272314 T^{10} + 3412018000 T^{11} + 1823445881 p T^{12} + 85579508576 T^{13} + 691428574449 T^{14} + 1760474326102 T^{15} + 12855128851858 T^{16} + 1760474326102 p T^{17} + 691428574449 p^{2} T^{18} + 85579508576 p^{3} T^{19} + 1823445881 p^{5} T^{20} + 3412018000 p^{5} T^{21} + 1147272314 p^{6} T^{22} + 109711108 p^{7} T^{23} + 34419993 p^{8} T^{24} + 2759804 p^{9} T^{25} + 811354 p^{10} T^{26} + 51436 p^{11} T^{27} + 14230 p^{12} T^{28} + 638 p^{13} T^{29} + 167 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
19 \( 1 - 362 T^{2} + 65537 T^{4} - 7894372 T^{6} + 710531740 T^{8} - 2678680452 p T^{10} + 3018445951436 T^{12} - 152297002340810 T^{14} + 6667558203982732 T^{16} - 257106513456336340 T^{18} + 8835236510273005542 T^{20} - \)\(27\!\cdots\!72\)\( T^{22} + \)\(76\!\cdots\!52\)\( T^{24} - \)\(19\!\cdots\!04\)\( T^{26} + \)\(45\!\cdots\!65\)\( T^{28} - \)\(98\!\cdots\!88\)\( T^{30} + \)\(19\!\cdots\!99\)\( T^{32} - \)\(98\!\cdots\!88\)\( p^{2} T^{34} + \)\(45\!\cdots\!65\)\( p^{4} T^{36} - \)\(19\!\cdots\!04\)\( p^{6} T^{38} + \)\(76\!\cdots\!52\)\( p^{8} T^{40} - \)\(27\!\cdots\!72\)\( p^{10} T^{42} + 8835236510273005542 p^{12} T^{44} - 257106513456336340 p^{14} T^{46} + 6667558203982732 p^{16} T^{48} - 152297002340810 p^{18} T^{50} + 3018445951436 p^{20} T^{52} - 2678680452 p^{23} T^{54} + 710531740 p^{24} T^{56} - 7894372 p^{26} T^{58} + 65537 p^{28} T^{60} - 362 p^{30} T^{62} + p^{32} T^{64} \)
23 \( ( 1 + 4 T + 182 T^{2} + 876 T^{3} + 734 p T^{4} + 85596 T^{5} + 1066996 T^{6} + 5179642 T^{7} + 50683499 T^{8} + 225557226 T^{9} + 1897105692 T^{10} + 7709902006 T^{11} + 58547822885 T^{12} + 219399866216 T^{13} + 1565541834150 T^{14} + 5489823491962 T^{15} + 37688475152030 T^{16} + 5489823491962 p T^{17} + 1565541834150 p^{2} T^{18} + 219399866216 p^{3} T^{19} + 58547822885 p^{4} T^{20} + 7709902006 p^{5} T^{21} + 1897105692 p^{6} T^{22} + 225557226 p^{7} T^{23} + 50683499 p^{8} T^{24} + 5179642 p^{9} T^{25} + 1066996 p^{10} T^{26} + 85596 p^{11} T^{27} + 734 p^{13} T^{28} + 876 p^{13} T^{29} + 182 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
29 \( ( 1 + 8 T + 220 T^{2} + 1110 T^{3} + 21310 T^{4} + 68526 T^{5} + 1342857 T^{6} + 2179216 T^{7} + 63203332 T^{8} - 1461056 T^{9} + 2417825971 T^{10} - 4095947054 T^{11} + 80860738337 T^{12} - 256714762740 T^{13} + 2493206148736 T^{14} - 10114188632182 T^{15} + 73571341464604 T^{16} - 10114188632182 p T^{17} + 2493206148736 p^{2} T^{18} - 256714762740 p^{3} T^{19} + 80860738337 p^{4} T^{20} - 4095947054 p^{5} T^{21} + 2417825971 p^{6} T^{22} - 1461056 p^{7} T^{23} + 63203332 p^{8} T^{24} + 2179216 p^{9} T^{25} + 1342857 p^{10} T^{26} + 68526 p^{11} T^{27} + 21310 p^{12} T^{28} + 1110 p^{13} T^{29} + 220 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
37 \( 1 - 306 T^{2} + 52239 T^{4} - 6266872 T^{6} + 586269565 T^{8} - 45149545554 T^{10} + 2974098363097 T^{12} - 172481523313070 T^{14} + 9041234926960045 T^{16} - 438726695506207386 T^{18} + 20145535643000796629 T^{20} - \)\(88\!\cdots\!26\)\( T^{22} + \)\(38\!\cdots\!36\)\( T^{24} - \)\(15\!\cdots\!68\)\( T^{26} + \)\(64\!\cdots\!99\)\( T^{28} - \)\(25\!\cdots\!66\)\( T^{30} + \)\(95\!\cdots\!30\)\( T^{32} - \)\(25\!\cdots\!66\)\( p^{2} T^{34} + \)\(64\!\cdots\!99\)\( p^{4} T^{36} - \)\(15\!\cdots\!68\)\( p^{6} T^{38} + \)\(38\!\cdots\!36\)\( p^{8} T^{40} - \)\(88\!\cdots\!26\)\( p^{10} T^{42} + 20145535643000796629 p^{12} T^{44} - 438726695506207386 p^{14} T^{46} + 9041234926960045 p^{16} T^{48} - 172481523313070 p^{18} T^{50} + 2974098363097 p^{20} T^{52} - 45149545554 p^{22} T^{54} + 586269565 p^{24} T^{56} - 6266872 p^{26} T^{58} + 52239 p^{28} T^{60} - 306 p^{30} T^{62} + p^{32} T^{64} \)
41 \( 1 - 598 T^{2} + 177725 T^{4} - 34940964 T^{6} + 5105384150 T^{8} - 590932154992 T^{10} + 56451345060464 T^{12} - 4585231206792736 T^{14} + 324421659064064133 T^{16} - 498488805955044026 p T^{18} + \)\(11\!\cdots\!23\)\( T^{20} - \)\(62\!\cdots\!80\)\( T^{22} + \)\(31\!\cdots\!73\)\( T^{24} - \)\(15\!\cdots\!40\)\( T^{26} + \)\(69\!\cdots\!79\)\( T^{28} - \)\(30\!\cdots\!02\)\( T^{30} + \)\(31\!\cdots\!49\)\( p T^{32} - \)\(30\!\cdots\!02\)\( p^{2} T^{34} + \)\(69\!\cdots\!79\)\( p^{4} T^{36} - \)\(15\!\cdots\!40\)\( p^{6} T^{38} + \)\(31\!\cdots\!73\)\( p^{8} T^{40} - \)\(62\!\cdots\!80\)\( p^{10} T^{42} + \)\(11\!\cdots\!23\)\( p^{12} T^{44} - 498488805955044026 p^{15} T^{46} + 324421659064064133 p^{16} T^{48} - 4585231206792736 p^{18} T^{50} + 56451345060464 p^{20} T^{52} - 590932154992 p^{22} T^{54} + 5105384150 p^{24} T^{56} - 34940964 p^{26} T^{58} + 177725 p^{28} T^{60} - 598 p^{30} T^{62} + p^{32} T^{64} \)
43 \( ( 1 + 16 T + 491 T^{2} + 6320 T^{3} + 2631 p T^{4} + 1240932 T^{5} + 16691615 T^{6} + 161301798 T^{7} + 1792302122 T^{8} + 15573445800 T^{9} + 149872010422 T^{10} + 1185239906190 T^{11} + 10146060020032 T^{12} + 73496852836876 T^{13} + 568551289690556 T^{14} + 3780355123235520 T^{15} + 26672834634296236 T^{16} + 3780355123235520 p T^{17} + 568551289690556 p^{2} T^{18} + 73496852836876 p^{3} T^{19} + 10146060020032 p^{4} T^{20} + 1185239906190 p^{5} T^{21} + 149872010422 p^{6} T^{22} + 15573445800 p^{7} T^{23} + 1792302122 p^{8} T^{24} + 161301798 p^{9} T^{25} + 16691615 p^{10} T^{26} + 1240932 p^{11} T^{27} + 2631 p^{13} T^{28} + 6320 p^{13} T^{29} + 491 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
47 \( 1 - 538 T^{2} + 148995 T^{4} - 28373348 T^{6} + 4186047955 T^{8} - 510696684258 T^{10} + 53645530269145 T^{12} - 4984015306762692 T^{14} + 417287597760764624 T^{16} - 31914330297198507072 T^{18} + \)\(22\!\cdots\!08\)\( T^{20} - \)\(14\!\cdots\!56\)\( T^{22} + \)\(90\!\cdots\!28\)\( T^{24} - \)\(52\!\cdots\!94\)\( T^{26} + \)\(28\!\cdots\!08\)\( T^{28} - \)\(14\!\cdots\!82\)\( T^{30} + \)\(70\!\cdots\!96\)\( T^{32} - \)\(14\!\cdots\!82\)\( p^{2} T^{34} + \)\(28\!\cdots\!08\)\( p^{4} T^{36} - \)\(52\!\cdots\!94\)\( p^{6} T^{38} + \)\(90\!\cdots\!28\)\( p^{8} T^{40} - \)\(14\!\cdots\!56\)\( p^{10} T^{42} + \)\(22\!\cdots\!08\)\( p^{12} T^{44} - 31914330297198507072 p^{14} T^{46} + 417287597760764624 p^{16} T^{48} - 4984015306762692 p^{18} T^{50} + 53645530269145 p^{20} T^{52} - 510696684258 p^{22} T^{54} + 4186047955 p^{24} T^{56} - 28373348 p^{26} T^{58} + 148995 p^{28} T^{60} - 538 p^{30} T^{62} + p^{32} T^{64} \)
53 \( ( 1 + 6 T + 469 T^{2} + 3382 T^{3} + 113080 T^{4} + 910192 T^{5} + 18630434 T^{6} + 157418214 T^{7} + 2337521583 T^{8} + 19768857434 T^{9} + 234972166028 T^{10} + 1922908429030 T^{11} + 19415354023723 T^{12} + 150287264202844 T^{13} + 1336877045345107 T^{14} + 9629706833693278 T^{15} + 77256050069348266 T^{16} + 9629706833693278 p T^{17} + 1336877045345107 p^{2} T^{18} + 150287264202844 p^{3} T^{19} + 19415354023723 p^{4} T^{20} + 1922908429030 p^{5} T^{21} + 234972166028 p^{6} T^{22} + 19768857434 p^{7} T^{23} + 2337521583 p^{8} T^{24} + 157418214 p^{9} T^{25} + 18630434 p^{10} T^{26} + 910192 p^{11} T^{27} + 113080 p^{12} T^{28} + 3382 p^{13} T^{29} + 469 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
59 \( 1 - 618 T^{2} + 209343 T^{4} - 50550328 T^{6} + 9655450533 T^{8} - 1541684376740 T^{10} + 212881093721401 T^{12} - 26006607931665120 T^{14} + 48418719899207291 p T^{16} - \)\(28\!\cdots\!28\)\( T^{18} + \)\(26\!\cdots\!90\)\( T^{20} - \)\(22\!\cdots\!40\)\( T^{22} + \)\(17\!\cdots\!87\)\( T^{24} - \)\(21\!\cdots\!14\)\( p T^{26} + \)\(89\!\cdots\!98\)\( T^{28} - \)\(57\!\cdots\!88\)\( T^{30} + \)\(35\!\cdots\!81\)\( T^{32} - \)\(57\!\cdots\!88\)\( p^{2} T^{34} + \)\(89\!\cdots\!98\)\( p^{4} T^{36} - \)\(21\!\cdots\!14\)\( p^{7} T^{38} + \)\(17\!\cdots\!87\)\( p^{8} T^{40} - \)\(22\!\cdots\!40\)\( p^{10} T^{42} + \)\(26\!\cdots\!90\)\( p^{12} T^{44} - \)\(28\!\cdots\!28\)\( p^{14} T^{46} + 48418719899207291 p^{17} T^{48} - 26006607931665120 p^{18} T^{50} + 212881093721401 p^{20} T^{52} - 1541684376740 p^{22} T^{54} + 9655450533 p^{24} T^{56} - 50550328 p^{26} T^{58} + 209343 p^{28} T^{60} - 618 p^{30} T^{62} + p^{32} T^{64} \)
61 \( ( 1 - 8 T + 605 T^{2} - 4160 T^{3} + 179090 T^{4} - 1077090 T^{5} + 34658063 T^{6} - 184200124 T^{7} + 4937753413 T^{8} - 23384070246 T^{9} + 552964363604 T^{10} - 2353709635954 T^{11} + 50716482075547 T^{12} - 195982769600702 T^{13} + 3911640281431720 T^{14} - 13871953720143360 T^{15} + 257587502471292710 T^{16} - 13871953720143360 p T^{17} + 3911640281431720 p^{2} T^{18} - 195982769600702 p^{3} T^{19} + 50716482075547 p^{4} T^{20} - 2353709635954 p^{5} T^{21} + 552964363604 p^{6} T^{22} - 23384070246 p^{7} T^{23} + 4937753413 p^{8} T^{24} - 184200124 p^{9} T^{25} + 34658063 p^{10} T^{26} - 1077090 p^{11} T^{27} + 179090 p^{12} T^{28} - 4160 p^{13} T^{29} + 605 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
67 \( 1 - 1016 T^{2} + 527832 T^{4} - 186497968 T^{6} + 50296699668 T^{8} - 11019051359946 T^{10} + 2038379247870750 T^{12} - 326797593127146610 T^{14} + 46254522077304320772 T^{16} - \)\(58\!\cdots\!98\)\( T^{18} + \)\(67\!\cdots\!39\)\( T^{20} - \)\(69\!\cdots\!92\)\( T^{22} + \)\(66\!\cdots\!19\)\( T^{24} - \)\(58\!\cdots\!42\)\( T^{26} + \)\(47\!\cdots\!63\)\( T^{28} - \)\(35\!\cdots\!20\)\( T^{30} + \)\(24\!\cdots\!44\)\( T^{32} - \)\(35\!\cdots\!20\)\( p^{2} T^{34} + \)\(47\!\cdots\!63\)\( p^{4} T^{36} - \)\(58\!\cdots\!42\)\( p^{6} T^{38} + \)\(66\!\cdots\!19\)\( p^{8} T^{40} - \)\(69\!\cdots\!92\)\( p^{10} T^{42} + \)\(67\!\cdots\!39\)\( p^{12} T^{44} - \)\(58\!\cdots\!98\)\( p^{14} T^{46} + 46254522077304320772 p^{16} T^{48} - 326797593127146610 p^{18} T^{50} + 2038379247870750 p^{20} T^{52} - 11019051359946 p^{22} T^{54} + 50296699668 p^{24} T^{56} - 186497968 p^{26} T^{58} + 527832 p^{28} T^{60} - 1016 p^{30} T^{62} + p^{32} T^{64} \)
71 \( 1 - 1546 T^{2} + 1183685 T^{4} - 597928620 T^{6} + 223987371164 T^{8} - 66313061033388 T^{10} + 16147411618395400 T^{12} - 3323144240356609954 T^{14} + \)\(58\!\cdots\!48\)\( T^{16} - \)\(91\!\cdots\!00\)\( T^{18} + \)\(12\!\cdots\!62\)\( T^{20} - \)\(15\!\cdots\!60\)\( T^{22} + \)\(16\!\cdots\!24\)\( T^{24} - \)\(16\!\cdots\!80\)\( T^{26} + \)\(15\!\cdots\!93\)\( T^{28} - \)\(12\!\cdots\!52\)\( T^{30} + \)\(91\!\cdots\!35\)\( T^{32} - \)\(12\!\cdots\!52\)\( p^{2} T^{34} + \)\(15\!\cdots\!93\)\( p^{4} T^{36} - \)\(16\!\cdots\!80\)\( p^{6} T^{38} + \)\(16\!\cdots\!24\)\( p^{8} T^{40} - \)\(15\!\cdots\!60\)\( p^{10} T^{42} + \)\(12\!\cdots\!62\)\( p^{12} T^{44} - \)\(91\!\cdots\!00\)\( p^{14} T^{46} + \)\(58\!\cdots\!48\)\( p^{16} T^{48} - 3323144240356609954 p^{18} T^{50} + 16147411618395400 p^{20} T^{52} - 66313061033388 p^{22} T^{54} + 223987371164 p^{24} T^{56} - 597928620 p^{26} T^{58} + 1183685 p^{28} T^{60} - 1546 p^{30} T^{62} + p^{32} T^{64} \)
73 \( 1 - 1260 T^{2} + 786012 T^{4} - 324310302 T^{6} + 99778363228 T^{8} - 24469776877702 T^{10} + 4993107147824523 T^{12} - 873501848000267116 T^{14} + \)\(13\!\cdots\!10\)\( T^{16} - \)\(18\!\cdots\!54\)\( T^{18} + \)\(22\!\cdots\!55\)\( T^{20} - \)\(25\!\cdots\!00\)\( T^{22} + \)\(25\!\cdots\!05\)\( T^{24} - \)\(24\!\cdots\!04\)\( T^{26} + \)\(21\!\cdots\!14\)\( T^{28} - \)\(17\!\cdots\!62\)\( T^{30} + \)\(13\!\cdots\!36\)\( T^{32} - \)\(17\!\cdots\!62\)\( p^{2} T^{34} + \)\(21\!\cdots\!14\)\( p^{4} T^{36} - \)\(24\!\cdots\!04\)\( p^{6} T^{38} + \)\(25\!\cdots\!05\)\( p^{8} T^{40} - \)\(25\!\cdots\!00\)\( p^{10} T^{42} + \)\(22\!\cdots\!55\)\( p^{12} T^{44} - \)\(18\!\cdots\!54\)\( p^{14} T^{46} + \)\(13\!\cdots\!10\)\( p^{16} T^{48} - 873501848000267116 p^{18} T^{50} + 4993107147824523 p^{20} T^{52} - 24469776877702 p^{22} T^{54} + 99778363228 p^{24} T^{56} - 324310302 p^{26} T^{58} + 786012 p^{28} T^{60} - 1260 p^{30} T^{62} + p^{32} T^{64} \)
79 \( ( 1 - 32 T + 1315 T^{2} - 30182 T^{3} + 9297 p T^{4} - 13343174 T^{5} + 243579549 T^{6} - 3692680710 T^{7} + 55185674243 T^{8} - 721646400906 T^{9} + 9239431689267 T^{10} - 106654598620084 T^{11} + 1203548318055452 T^{12} - 12472310267258742 T^{13} + 126429719218540843 T^{14} - 1189380926010068494 T^{15} + 10954443490430363522 T^{16} - 1189380926010068494 p T^{17} + 126429719218540843 p^{2} T^{18} - 12472310267258742 p^{3} T^{19} + 1203548318055452 p^{4} T^{20} - 106654598620084 p^{5} T^{21} + 9239431689267 p^{6} T^{22} - 721646400906 p^{7} T^{23} + 55185674243 p^{8} T^{24} - 3692680710 p^{9} T^{25} + 243579549 p^{10} T^{26} - 13343174 p^{11} T^{27} + 9297 p^{13} T^{28} - 30182 p^{13} T^{29} + 1315 p^{14} T^{30} - 32 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
83 \( 1 - 1234 T^{2} + 792423 T^{4} - 350287656 T^{6} + 119152153847 T^{8} - 33093861669950 T^{10} + 7783312719287805 T^{12} - 1588179924431712976 T^{14} + \)\(28\!\cdots\!04\)\( T^{16} - \)\(46\!\cdots\!40\)\( T^{18} + \)\(66\!\cdots\!60\)\( T^{20} - \)\(10\!\cdots\!24\)\( p T^{22} + \)\(10\!\cdots\!56\)\( T^{24} - \)\(11\!\cdots\!02\)\( T^{26} + \)\(11\!\cdots\!04\)\( T^{28} - \)\(11\!\cdots\!42\)\( T^{30} + \)\(96\!\cdots\!92\)\( T^{32} - \)\(11\!\cdots\!42\)\( p^{2} T^{34} + \)\(11\!\cdots\!04\)\( p^{4} T^{36} - \)\(11\!\cdots\!02\)\( p^{6} T^{38} + \)\(10\!\cdots\!56\)\( p^{8} T^{40} - \)\(10\!\cdots\!24\)\( p^{11} T^{42} + \)\(66\!\cdots\!60\)\( p^{12} T^{44} - \)\(46\!\cdots\!40\)\( p^{14} T^{46} + \)\(28\!\cdots\!04\)\( p^{16} T^{48} - 1588179924431712976 p^{18} T^{50} + 7783312719287805 p^{20} T^{52} - 33093861669950 p^{22} T^{54} + 119152153847 p^{24} T^{56} - 350287656 p^{26} T^{58} + 792423 p^{28} T^{60} - 1234 p^{30} T^{62} + p^{32} T^{64} \)
89 \( 1 - 1762 T^{2} + 1522405 T^{4} - 859124854 T^{6} + 355976041976 T^{8} - 115485313170926 T^{10} + 30565656616309839 T^{12} - 6795771971847561776 T^{14} + \)\(12\!\cdots\!73\)\( T^{16} - \)\(21\!\cdots\!06\)\( T^{18} + \)\(32\!\cdots\!84\)\( T^{20} - \)\(43\!\cdots\!54\)\( T^{22} + \)\(53\!\cdots\!99\)\( T^{24} - \)\(60\!\cdots\!58\)\( T^{26} + \)\(63\!\cdots\!44\)\( T^{28} - \)\(62\!\cdots\!44\)\( T^{30} + \)\(57\!\cdots\!82\)\( T^{32} - \)\(62\!\cdots\!44\)\( p^{2} T^{34} + \)\(63\!\cdots\!44\)\( p^{4} T^{36} - \)\(60\!\cdots\!58\)\( p^{6} T^{38} + \)\(53\!\cdots\!99\)\( p^{8} T^{40} - \)\(43\!\cdots\!54\)\( p^{10} T^{42} + \)\(32\!\cdots\!84\)\( p^{12} T^{44} - \)\(21\!\cdots\!06\)\( p^{14} T^{46} + \)\(12\!\cdots\!73\)\( p^{16} T^{48} - 6795771971847561776 p^{18} T^{50} + 30565656616309839 p^{20} T^{52} - 115485313170926 p^{22} T^{54} + 355976041976 p^{24} T^{56} - 859124854 p^{26} T^{58} + 1522405 p^{28} T^{60} - 1762 p^{30} T^{62} + p^{32} T^{64} \)
97 \( 1 - 1410 T^{2} + 1001815 T^{4} - 476299434 T^{6} + 169921169505 T^{8} - 48396516374204 T^{10} + 11441797038082036 T^{12} - 2306907946781925514 T^{14} + \)\(40\!\cdots\!76\)\( T^{16} - \)\(62\!\cdots\!50\)\( T^{18} + \)\(87\!\cdots\!40\)\( T^{20} - \)\(11\!\cdots\!22\)\( T^{22} + \)\(13\!\cdots\!52\)\( T^{24} - \)\(14\!\cdots\!92\)\( T^{26} + \)\(15\!\cdots\!07\)\( T^{28} - \)\(15\!\cdots\!14\)\( T^{30} + \)\(14\!\cdots\!89\)\( T^{32} - \)\(15\!\cdots\!14\)\( p^{2} T^{34} + \)\(15\!\cdots\!07\)\( p^{4} T^{36} - \)\(14\!\cdots\!92\)\( p^{6} T^{38} + \)\(13\!\cdots\!52\)\( p^{8} T^{40} - \)\(11\!\cdots\!22\)\( p^{10} T^{42} + \)\(87\!\cdots\!40\)\( p^{12} T^{44} - \)\(62\!\cdots\!50\)\( p^{14} T^{46} + \)\(40\!\cdots\!76\)\( p^{16} T^{48} - 2306907946781925514 p^{18} T^{50} + 11441797038082036 p^{20} T^{52} - 48396516374204 p^{22} T^{54} + 169921169505 p^{24} T^{56} - 476299434 p^{26} T^{58} + 1001815 p^{28} T^{60} - 1410 p^{30} T^{62} + p^{32} T^{64} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{64} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.06046572575067476436516906155, −2.05450961566423557475382835359, −2.03367276864621741948266295624, −1.98158137384099356205838194186, −1.96408967457189775180686703842, −1.83376102809879966433373630506, −1.79338321221897472760009807780, −1.76924368451921999607736763013, −1.47240629001336519367520001298, −1.31477744630168698078845831534, −1.30118858579066313038652041047, −1.22977579475967340979005607855, −1.17199129180240821136556035084, −1.16058977129461835611252743019, −1.08251451617496521826429578685, −1.04088348207883971394057690950, −0.939047223363470004914495963438, −0.875967632115337701482663071860, −0.75473608244935910638462840413, −0.74302558891654016651960838591, −0.63914791125131542572731536242, −0.54772789827394013181865708734, −0.41489555450784126834565589529, −0.31724277709481984017687016418, −0.25804067569286527232360389081, 0.25804067569286527232360389081, 0.31724277709481984017687016418, 0.41489555450784126834565589529, 0.54772789827394013181865708734, 0.63914791125131542572731536242, 0.74302558891654016651960838591, 0.75473608244935910638462840413, 0.875967632115337701482663071860, 0.939047223363470004914495963438, 1.04088348207883971394057690950, 1.08251451617496521826429578685, 1.16058977129461835611252743019, 1.17199129180240821136556035084, 1.22977579475967340979005607855, 1.30118858579066313038652041047, 1.31477744630168698078845831534, 1.47240629001336519367520001298, 1.76924368451921999607736763013, 1.79338321221897472760009807780, 1.83376102809879966433373630506, 1.96408967457189775180686703842, 1.98158137384099356205838194186, 2.03367276864621741948266295624, 2.05450961566423557475382835359, 2.06046572575067476436516906155

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

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