Properties

Label 32-33e32-1.1-c2e16-0-3
Degree $32$
Conductor $3.912\times 10^{48}$
Sign $1$
Analytic cond. $3.61249\times 10^{23}$
Root an. cond. $5.44730$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 22·4-s + 4·5-s + 211·16-s + 88·20-s − 132·23-s − 148·25-s + 40·31-s − 16·37-s − 80·47-s + 322·49-s + 128·53-s + 220·59-s + 1.06e3·64-s + 36·67-s − 644·71-s + 844·80-s − 76·89-s − 2.90e3·92-s + 216·97-s − 3.25e3·100-s + 68·103-s − 332·113-s − 528·115-s + 880·124-s − 584·125-s + 127-s + 131-s + ⋯
L(s)  = 1  + 11/2·4-s + 4/5·5-s + 13.1·16-s + 22/5·20-s − 5.73·23-s − 5.91·25-s + 1.29·31-s − 0.432·37-s − 1.70·47-s + 46/7·49-s + 2.41·53-s + 3.72·59-s + 16.5·64-s + 0.537·67-s − 9.07·71-s + 10.5·80-s − 0.853·89-s − 31.5·92-s + 2.22·97-s − 32.5·100-s + 0.660·103-s − 2.93·113-s − 4.59·115-s + 7.09·124-s − 4.67·125-s + 0.00787·127-s + 0.00763·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 11^{32}\)
Sign: $1$
Analytic conductor: \(3.61249\times 10^{23}\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1089} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{32} \cdot 11^{32} ,\ ( \ : [1]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(24.09543314\)
\(L(\frac12)\) \(\approx\) \(24.09543314\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 - 11 p T^{2} + 273 T^{4} - 1213 p T^{6} + 4243 p^{2} T^{8} - 49415 p T^{10} + 499377 T^{12} - 1130941 p T^{14} + 9401145 T^{16} - 1130941 p^{5} T^{18} + 499377 p^{8} T^{20} - 49415 p^{13} T^{22} + 4243 p^{18} T^{24} - 1213 p^{21} T^{26} + 273 p^{24} T^{28} - 11 p^{29} T^{30} + p^{32} T^{32} \)
5 \( ( 1 - 2 T + 16 p T^{2} - 172 T^{3} + 3129 T^{4} - 5738 T^{5} + 18976 p T^{6} - 110568 T^{7} + 2554561 T^{8} - 110568 p^{2} T^{9} + 18976 p^{5} T^{10} - 5738 p^{6} T^{11} + 3129 p^{8} T^{12} - 172 p^{10} T^{13} + 16 p^{13} T^{14} - 2 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
7 \( 1 - 46 p T^{2} + 58341 T^{4} - 7573388 T^{6} + 771680206 T^{8} - 64750351786 T^{10} + 655863150672 p T^{12} - 279301051376894 T^{14} + 14697673922068797 T^{16} - 279301051376894 p^{4} T^{18} + 655863150672 p^{9} T^{20} - 64750351786 p^{12} T^{22} + 771680206 p^{16} T^{24} - 7573388 p^{20} T^{26} + 58341 p^{24} T^{28} - 46 p^{29} T^{30} + p^{32} T^{32} \)
13 \( 1 - 1588 T^{2} + 1282374 T^{4} - 693507560 T^{6} + 280155848113 T^{8} - 89505604820920 T^{10} + 23383361940906966 T^{12} - 5095739810118559532 T^{14} + \)\(93\!\cdots\!68\)\( T^{16} - 5095739810118559532 p^{4} T^{18} + 23383361940906966 p^{8} T^{20} - 89505604820920 p^{12} T^{22} + 280155848113 p^{16} T^{24} - 693507560 p^{20} T^{26} + 1282374 p^{24} T^{28} - 1588 p^{28} T^{30} + p^{32} T^{32} \)
17 \( 1 - 2630 T^{2} + 3599671 T^{4} - 3343874680 T^{6} + 2336603839655 T^{8} - 1294245135151940 T^{10} + 585438285841824869 T^{12} - \)\(22\!\cdots\!30\)\( T^{14} + \)\(69\!\cdots\!84\)\( T^{16} - \)\(22\!\cdots\!30\)\( p^{4} T^{18} + 585438285841824869 p^{8} T^{20} - 1294245135151940 p^{12} T^{22} + 2336603839655 p^{16} T^{24} - 3343874680 p^{20} T^{26} + 3599671 p^{24} T^{28} - 2630 p^{28} T^{30} + p^{32} T^{32} \)
19 \( 1 - 178 p T^{2} + 5627679 T^{4} - 6148910900 T^{6} + 4984788194863 T^{8} - 3212178561172840 T^{10} + 1715595160585663581 T^{12} - \)\(77\!\cdots\!38\)\( T^{14} + \)\(30\!\cdots\!88\)\( T^{16} - \)\(77\!\cdots\!38\)\( p^{4} T^{18} + 1715595160585663581 p^{8} T^{20} - 3212178561172840 p^{12} T^{22} + 4984788194863 p^{16} T^{24} - 6148910900 p^{20} T^{26} + 5627679 p^{24} T^{28} - 178 p^{29} T^{30} + p^{32} T^{32} \)
23 \( ( 1 + 66 T + 5094 T^{2} + 230356 T^{3} + 10182141 T^{4} + 348333808 T^{5} + 11089711126 T^{6} + 298981380378 T^{7} + 7385976715852 T^{8} + 298981380378 p^{2} T^{9} + 11089711126 p^{4} T^{10} + 348333808 p^{6} T^{11} + 10182141 p^{8} T^{12} + 230356 p^{10} T^{13} + 5094 p^{12} T^{14} + 66 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
29 \( 1 - 7642 T^{2} + 29638239 T^{4} - 77141788220 T^{6} + 150617121597103 T^{8} - 233913712905223000 T^{10} + \)\(29\!\cdots\!41\)\( T^{12} - \)\(32\!\cdots\!18\)\( T^{14} + \)\(29\!\cdots\!08\)\( T^{16} - \)\(32\!\cdots\!18\)\( p^{4} T^{18} + \)\(29\!\cdots\!41\)\( p^{8} T^{20} - 233913712905223000 p^{12} T^{22} + 150617121597103 p^{16} T^{24} - 77141788220 p^{20} T^{26} + 29638239 p^{24} T^{28} - 7642 p^{28} T^{30} + p^{32} T^{32} \)
31 \( ( 1 - 20 T + 3590 T^{2} - 99880 T^{3} + 6923319 T^{4} - 231217460 T^{5} + 9449704220 T^{6} - 329324583720 T^{7} + 10065520246261 T^{8} - 329324583720 p^{2} T^{9} + 9449704220 p^{4} T^{10} - 231217460 p^{6} T^{11} + 6923319 p^{8} T^{12} - 99880 p^{10} T^{13} + 3590 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
37 \( ( 1 + 8 T + 4642 T^{2} - 31908 T^{3} + 9775637 T^{4} - 196180240 T^{5} + 17813523498 T^{6} - 359847378724 T^{7} + 28928176861580 T^{8} - 359847378724 p^{2} T^{9} + 17813523498 p^{4} T^{10} - 196180240 p^{6} T^{11} + 9775637 p^{8} T^{12} - 31908 p^{10} T^{13} + 4642 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
41 \( 1 - 15440 T^{2} + 122127646 T^{4} - 652897746400 T^{6} + 2625127943811665 T^{8} - 8378945095931777600 T^{10} + \)\(21\!\cdots\!54\)\( T^{12} - \)\(47\!\cdots\!40\)\( T^{14} + \)\(87\!\cdots\!84\)\( T^{16} - \)\(47\!\cdots\!40\)\( p^{4} T^{18} + \)\(21\!\cdots\!54\)\( p^{8} T^{20} - 8378945095931777600 p^{12} T^{22} + 2625127943811665 p^{16} T^{24} - 652897746400 p^{20} T^{26} + 122127646 p^{24} T^{28} - 15440 p^{28} T^{30} + p^{32} T^{32} \)
43 \( 1 - 12876 T^{2} + 91501662 T^{4} - 458248399064 T^{6} + 1782172921950441 T^{8} - 5666821055429722520 T^{10} + \)\(15\!\cdots\!78\)\( T^{12} - \)\(34\!\cdots\!80\)\( T^{14} + \)\(69\!\cdots\!12\)\( T^{16} - \)\(34\!\cdots\!80\)\( p^{4} T^{18} + \)\(15\!\cdots\!78\)\( p^{8} T^{20} - 5666821055429722520 p^{12} T^{22} + 1782172921950441 p^{16} T^{24} - 458248399064 p^{20} T^{26} + 91501662 p^{24} T^{28} - 12876 p^{28} T^{30} + p^{32} T^{32} \)
47 \( ( 1 + 40 T + 12889 T^{2} + 473990 T^{3} + 79513175 T^{4} + 2565920890 T^{5} + 305649430451 T^{6} + 8484233092760 T^{7} + 804587964496504 T^{8} + 8484233092760 p^{2} T^{9} + 305649430451 p^{4} T^{10} + 2565920890 p^{6} T^{11} + 79513175 p^{8} T^{12} + 473990 p^{10} T^{13} + 12889 p^{12} T^{14} + 40 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
53 \( ( 1 - 64 T + 7072 T^{2} - 155958 T^{3} + 16770977 T^{4} - 78338920 T^{5} + 34035352248 T^{6} + 801754600304 T^{7} + 51163449942665 T^{8} + 801754600304 p^{2} T^{9} + 34035352248 p^{4} T^{10} - 78338920 p^{6} T^{11} + 16770977 p^{8} T^{12} - 155958 p^{10} T^{13} + 7072 p^{12} T^{14} - 64 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
59 \( ( 1 - 110 T + 22460 T^{2} - 1924240 T^{3} + 221965089 T^{4} - 15515387750 T^{5} + 1323755544860 T^{6} - 77758052407680 T^{7} + 5435950311634261 T^{8} - 77758052407680 p^{2} T^{9} + 1323755544860 p^{4} T^{10} - 15515387750 p^{6} T^{11} + 221965089 p^{8} T^{12} - 1924240 p^{10} T^{13} + 22460 p^{12} T^{14} - 110 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
61 \( 1 - 32230 T^{2} + 486184863 T^{4} - 4498448685020 T^{6} + 27891920151604423 T^{8} - 1918041474980927080 p T^{10} + \)\(30\!\cdots\!81\)\( T^{12} - \)\(31\!\cdots\!50\)\( T^{14} - \)\(33\!\cdots\!60\)\( T^{16} - \)\(31\!\cdots\!50\)\( p^{4} T^{18} + \)\(30\!\cdots\!81\)\( p^{8} T^{20} - 1918041474980927080 p^{13} T^{22} + 27891920151604423 p^{16} T^{24} - 4498448685020 p^{20} T^{26} + 486184863 p^{24} T^{28} - 32230 p^{28} T^{30} + p^{32} T^{32} \)
67 \( ( 1 - 18 T + 21975 T^{2} - 668180 T^{3} + 239231415 T^{4} - 8253401324 T^{5} + 1771382320897 T^{6} - 55179990303630 T^{7} + 9425465643780880 T^{8} - 55179990303630 p^{2} T^{9} + 1771382320897 p^{4} T^{10} - 8253401324 p^{6} T^{11} + 239231415 p^{8} T^{12} - 668180 p^{10} T^{13} + 21975 p^{12} T^{14} - 18 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
71 \( ( 1 + 322 T + 75709 T^{2} + 12666110 T^{3} + 1765380563 T^{4} + 203089187710 T^{5} + 20322138766751 T^{6} + 1751047193368898 T^{7} + 133272409761269728 T^{8} + 1751047193368898 p^{2} T^{9} + 20322138766751 p^{4} T^{10} + 203089187710 p^{6} T^{11} + 1765380563 p^{8} T^{12} + 12666110 p^{10} T^{13} + 75709 p^{12} T^{14} + 322 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
73 \( 1 - 55862 T^{2} + 1492266943 T^{4} - 25450459258996 T^{6} + 312673246226141807 T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(22\!\cdots\!17\)\( T^{12} - \)\(14\!\cdots\!62\)\( T^{14} + \)\(84\!\cdots\!00\)\( T^{16} - \)\(14\!\cdots\!62\)\( p^{4} T^{18} + \)\(22\!\cdots\!17\)\( p^{8} T^{20} - \)\(29\!\cdots\!00\)\( p^{12} T^{22} + 312673246226141807 p^{16} T^{24} - 25450459258996 p^{20} T^{26} + 1492266943 p^{24} T^{28} - 55862 p^{28} T^{30} + p^{32} T^{32} \)
79 \( 1 - 63154 T^{2} + 1999538277 T^{4} - 42011760326036 T^{6} + 654243422431659166 T^{8} - 1280802837621484570 p^{2} T^{10} + \)\(79\!\cdots\!48\)\( T^{12} - \)\(64\!\cdots\!30\)\( T^{14} + \)\(44\!\cdots\!17\)\( T^{16} - \)\(64\!\cdots\!30\)\( p^{4} T^{18} + \)\(79\!\cdots\!48\)\( p^{8} T^{20} - 1280802837621484570 p^{14} T^{22} + 654243422431659166 p^{16} T^{24} - 42011760326036 p^{20} T^{26} + 1999538277 p^{24} T^{28} - 63154 p^{28} T^{30} + p^{32} T^{32} \)
83 \( 1 - 32498 T^{2} + 478780669 T^{4} - 5250389172760 T^{6} + 56791909914234278 T^{8} - \)\(54\!\cdots\!50\)\( T^{10} + \)\(45\!\cdots\!56\)\( T^{12} - \)\(35\!\cdots\!42\)\( T^{14} + \)\(26\!\cdots\!53\)\( T^{16} - \)\(35\!\cdots\!42\)\( p^{4} T^{18} + \)\(45\!\cdots\!56\)\( p^{8} T^{20} - \)\(54\!\cdots\!50\)\( p^{12} T^{22} + 56791909914234278 p^{16} T^{24} - 5250389172760 p^{20} T^{26} + 478780669 p^{24} T^{28} - 32498 p^{28} T^{30} + p^{32} T^{32} \)
89 \( ( 1 + 38 T + 37118 T^{2} + 1566244 T^{3} + 705775677 T^{4} + 28974031040 T^{5} + 8919242373542 T^{6} + 334883278291638 T^{7} + 81740346306794380 T^{8} + 334883278291638 p^{2} T^{9} + 8919242373542 p^{4} T^{10} + 28974031040 p^{6} T^{11} + 705775677 p^{8} T^{12} + 1566244 p^{10} T^{13} + 37118 p^{12} T^{14} + 38 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
97 \( ( 1 - 108 T + 60602 T^{2} - 6316692 T^{3} + 1715489247 T^{4} - 164951790420 T^{5} + 29502858551128 T^{6} - 2488741887330816 T^{7} + 336781879342073585 T^{8} - 2488741887330816 p^{2} T^{9} + 29502858551128 p^{4} T^{10} - 164951790420 p^{6} T^{11} + 1715489247 p^{8} T^{12} - 6316692 p^{10} T^{13} + 60602 p^{12} T^{14} - 108 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.28758637373618652949034641019, −2.24907560773612888953570509197, −2.04614038432663936773178679458, −1.97112130557277363050711723959, −1.89540056603917743225799742406, −1.88899772931897919518561366811, −1.88602200302502426406229143005, −1.83015682245037311022683633693, −1.79517740283026464992375387870, −1.72035599350717429639966849250, −1.69240746883897660600328481376, −1.57897356956707809081001798807, −1.53031276748863589147115362284, −1.49240723131839585819411956239, −1.20667322360646918193004472680, −1.01406075616210308193756737431, −0.975170209889002783990666921105, −0.792332597461235407280073036737, −0.77486896536150257290862506551, −0.71413323541313740925684957657, −0.43259005494844178760100555540, −0.40633361313443923992498740497, −0.40570667921924679923679614869, −0.13441836168448476663071239206, −0.10833044211374148545956943688, 0.10833044211374148545956943688, 0.13441836168448476663071239206, 0.40570667921924679923679614869, 0.40633361313443923992498740497, 0.43259005494844178760100555540, 0.71413323541313740925684957657, 0.77486896536150257290862506551, 0.792332597461235407280073036737, 0.975170209889002783990666921105, 1.01406075616210308193756737431, 1.20667322360646918193004472680, 1.49240723131839585819411956239, 1.53031276748863589147115362284, 1.57897356956707809081001798807, 1.69240746883897660600328481376, 1.72035599350717429639966849250, 1.79517740283026464992375387870, 1.83015682245037311022683633693, 1.88602200302502426406229143005, 1.88899772931897919518561366811, 1.89540056603917743225799742406, 1.97112130557277363050711723959, 2.04614038432663936773178679458, 2.24907560773612888953570509197, 2.28758637373618652949034641019

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

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