Dirichlet series
| L(s) = 1 | + 16·2-s + 136·4-s + 2·5-s + 816·8-s − 8·9-s + 32·10-s + 2·11-s + 3.87e3·16-s + 38·17-s − 128·18-s + 272·20-s + 32·22-s + 20·23-s + 1.55e4·32-s + 608·34-s − 1.08e3·36-s + 4·37-s + 1.63e3·40-s − 6·41-s + 2·43-s + 272·44-s − 16·45-s + 320·46-s + 47·49-s + 4·55-s + 5.42e4·64-s + 5.16e3·68-s + ⋯ |
| L(s) = 1 | + 11.3·2-s + 68·4-s + 0.894·5-s + 288.·8-s − 8/3·9-s + 10.1·10-s + 0.603·11-s + 969·16-s + 9.21·17-s − 30.1·18-s + 60.8·20-s + 6.82·22-s + 4.17·23-s + 2.74e3·32-s + 104.·34-s − 181.·36-s + 0.657·37-s + 258.·40-s − 0.937·41-s + 0.304·43-s + 41.0·44-s − 2.38·45-s + 47.1·46-s + 47/7·49-s + 0.539·55-s + 6.78e3·64-s + 626.·68-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 37^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 37^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 37^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.45077\times 10^{15}\) |
| Root analytic conductor: | \(2.97714\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 37^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(262413.5694\) |
| \(L(\frac12)\) | \(\approx\) | \(262413.5694\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{16} \) |
| 3 | \( ( 1 + T^{2} )^{8} \) | |
| 5 | \( 1 - 2 T + 4 T^{2} - 14 T^{3} - 8 T^{4} - 42 T^{5} - 36 T^{6} + 554 T^{7} - 338 T^{8} + 554 p T^{9} - 36 p^{2} T^{10} - 42 p^{3} T^{11} - 8 p^{4} T^{12} - 14 p^{5} T^{13} + 4 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 4 T - 64 T^{2} + 676 T^{3} + 28 T^{4} - 34500 T^{5} + 181312 T^{6} + 572100 T^{7} - 10388826 T^{8} + 572100 p T^{9} + 181312 p^{2} T^{10} - 34500 p^{3} T^{11} + 28 p^{4} T^{12} + 676 p^{5} T^{13} - 64 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 7 | \( 1 - 47 T^{2} + 156 p T^{4} - 2277 p T^{6} + 153245 T^{8} - 123632 p T^{10} + 310370 T^{12} + 46739074 T^{14} - 68892888 p T^{16} + 46739074 p^{2} T^{18} + 310370 p^{4} T^{20} - 123632 p^{7} T^{22} + 153245 p^{8} T^{24} - 2277 p^{11} T^{26} + 156 p^{13} T^{28} - 47 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 - T + 38 T^{2} - 5 p T^{3} + 625 T^{4} - 1704 T^{5} + 6102 T^{6} - 31558 T^{7} + 55584 T^{8} - 31558 p T^{9} + 6102 p^{2} T^{10} - 1704 p^{3} T^{11} + 625 p^{4} T^{12} - 5 p^{6} T^{13} + 38 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 70 T^{2} + 2365 T^{4} - 16 T^{5} + 51062 T^{6} - 704 T^{7} + 778764 T^{8} - 704 p T^{9} + 51062 p^{2} T^{10} - 16 p^{3} T^{11} + 2365 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( ( 1 - 19 T + 252 T^{2} - 2399 T^{3} + 18985 T^{4} - 124924 T^{5} + 716658 T^{6} - 3562754 T^{7} + 15697648 T^{8} - 3562754 p T^{9} + 716658 p^{2} T^{10} - 124924 p^{3} T^{11} + 18985 p^{4} T^{12} - 2399 p^{5} T^{13} + 252 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 148 T^{2} + 10974 T^{4} - 536680 T^{6} + 19336425 T^{8} - 548094968 T^{10} + 679696426 p T^{12} - 268317875868 T^{14} + 5206773948020 T^{16} - 268317875868 p^{2} T^{18} + 679696426 p^{5} T^{20} - 548094968 p^{6} T^{22} + 19336425 p^{8} T^{24} - 536680 p^{10} T^{26} + 10974 p^{12} T^{28} - 148 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 10 T + 142 T^{2} - 1042 T^{3} + 9137 T^{4} - 53168 T^{5} + 356342 T^{6} - 1733568 T^{7} + 9651700 T^{8} - 1733568 p T^{9} + 356342 p^{2} T^{10} - 53168 p^{3} T^{11} + 9137 p^{4} T^{12} - 1042 p^{5} T^{13} + 142 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 271 T^{2} + 37482 T^{4} - 3488373 T^{6} + 243288592 T^{8} - 13437261535 T^{10} + 606575631990 T^{12} - 22793116704493 T^{14} + 719896208733342 T^{16} - 22793116704493 p^{2} T^{18} + 606575631990 p^{4} T^{20} - 13437261535 p^{6} T^{22} + 243288592 p^{8} T^{24} - 3488373 p^{10} T^{26} + 37482 p^{12} T^{28} - 271 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 275 T^{2} + 38682 T^{4} - 3682017 T^{6} + 264557008 T^{8} - 15171501483 T^{10} + 716851848134 T^{12} - 28430445745121 T^{14} + 955791592071838 T^{16} - 28430445745121 p^{2} T^{18} + 716851848134 p^{4} T^{20} - 15171501483 p^{6} T^{22} + 264557008 p^{8} T^{24} - 3682017 p^{10} T^{26} + 38682 p^{12} T^{28} - 275 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 + 3 T + 130 T^{2} + 1049 T^{3} + 10764 T^{4} + 93079 T^{5} + 791494 T^{6} + 5004157 T^{7} + 39654230 T^{8} + 5004157 p T^{9} + 791494 p^{2} T^{10} + 93079 p^{3} T^{11} + 10764 p^{4} T^{12} + 1049 p^{5} T^{13} + 130 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - T + 134 T^{2} + 71 T^{3} + 11600 T^{4} + 10959 T^{5} + 708874 T^{6} + 902399 T^{7} + 33955262 T^{8} + 902399 p T^{9} + 708874 p^{2} T^{10} + 10959 p^{3} T^{11} + 11600 p^{4} T^{12} + 71 p^{5} T^{13} + 134 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 468 T^{2} + 108328 T^{4} - 16567308 T^{6} + 1881122860 T^{8} - 168435935396 T^{10} + 12307702920792 T^{12} - 748588721898876 T^{14} + 38298852492693414 T^{16} - 748588721898876 p^{2} T^{18} + 12307702920792 p^{4} T^{20} - 168435935396 p^{6} T^{22} + 1881122860 p^{8} T^{24} - 16567308 p^{10} T^{26} + 108328 p^{12} T^{28} - 468 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 431 T^{2} + 87208 T^{4} - 11082915 T^{6} + 1014484357 T^{8} - 74389247840 T^{10} + 4800154102634 T^{12} - 287276257242902 T^{14} + 15916111198541424 T^{16} - 287276257242902 p^{2} T^{18} + 4800154102634 p^{4} T^{20} - 74389247840 p^{6} T^{22} + 1014484357 p^{8} T^{24} - 11082915 p^{10} T^{26} + 87208 p^{12} T^{28} - 431 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 224 T^{2} + 30304 T^{4} - 3091648 T^{6} + 4456820 p T^{8} - 336765184 p T^{10} + 1395108692128 T^{12} - 91253367470176 T^{14} + 5579811466794566 T^{16} - 91253367470176 p^{2} T^{18} + 1395108692128 p^{4} T^{20} - 336765184 p^{7} T^{22} + 4456820 p^{9} T^{24} - 3091648 p^{10} T^{26} + 30304 p^{12} T^{28} - 224 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 - 595 T^{2} + 177718 T^{4} - 35316105 T^{6} + 5222000984 T^{8} - 608883375099 T^{10} + 57888834371658 T^{12} - 4576655097917049 T^{14} + 304047326847731278 T^{16} - 4576655097917049 p^{2} T^{18} + 57888834371658 p^{4} T^{20} - 608883375099 p^{6} T^{22} + 5222000984 p^{8} T^{24} - 35316105 p^{10} T^{26} + 177718 p^{12} T^{28} - 595 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 - 636 T^{2} + 201840 T^{4} - 42594356 T^{6} + 6712791820 T^{8} - 839327876380 T^{10} + 86170560178640 T^{12} - 7408166493293172 T^{14} + 538788774588568742 T^{16} - 7408166493293172 p^{2} T^{18} + 86170560178640 p^{4} T^{20} - 839327876380 p^{6} T^{22} + 6712791820 p^{8} T^{24} - 42594356 p^{10} T^{26} + 201840 p^{12} T^{28} - 636 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 12 T + 376 T^{2} + 3476 T^{3} + 67744 T^{4} + 536852 T^{5} + 7967224 T^{6} + 54029324 T^{7} + 660643614 T^{8} + 54029324 p T^{9} + 7967224 p^{2} T^{10} + 536852 p^{3} T^{11} + 67744 p^{4} T^{12} + 3476 p^{5} T^{13} + 376 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 - 132 T^{2} + 30110 T^{4} - 3397800 T^{6} + 447301321 T^{8} - 43111397896 T^{10} + 4171533320638 T^{12} - 4682990159468 p T^{14} + 26608301506913652 T^{16} - 4682990159468 p^{3} T^{18} + 4171533320638 p^{4} T^{20} - 43111397896 p^{6} T^{22} + 447301321 p^{8} T^{24} - 3397800 p^{10} T^{26} + 30110 p^{12} T^{28} - 132 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( 1 - 580 T^{2} + 153560 T^{4} - 25001644 T^{6} + 2942570988 T^{8} - 294735138468 T^{10} + 28902005682984 T^{12} - 2776393359848812 T^{14} + 237768649478745382 T^{16} - 2776393359848812 p^{2} T^{18} + 28902005682984 p^{4} T^{20} - 294735138468 p^{6} T^{22} + 2942570988 p^{8} T^{24} - 25001644 p^{10} T^{26} + 153560 p^{12} T^{28} - 580 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( 1 - 624 T^{2} + 193454 T^{4} - 40199860 T^{6} + 6376267201 T^{8} - 832774369752 T^{10} + 93757530349926 T^{12} - 9305009049015812 T^{14} + 819529123775548148 T^{16} - 9305009049015812 p^{2} T^{18} + 93757530349926 p^{4} T^{20} - 832774369752 p^{6} T^{22} + 6376267201 p^{8} T^{24} - 40199860 p^{10} T^{26} + 193454 p^{12} T^{28} - 624 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 - 452 T^{2} + 125974 T^{4} - 25319960 T^{6} + 4149081505 T^{8} - 572711813528 T^{10} + 69108861998278 T^{12} - 7333538698360716 T^{14} + 693148001682156356 T^{16} - 7333538698360716 p^{2} T^{18} + 69108861998278 p^{4} T^{20} - 572711813528 p^{6} T^{22} + 4149081505 p^{8} T^{24} - 25319960 p^{10} T^{26} + 125974 p^{12} T^{28} - 452 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 + 19 T + 364 T^{2} + 5517 T^{3} + 73844 T^{4} + 883587 T^{5} + 10147444 T^{6} + 107271981 T^{7} + 1093031078 T^{8} + 107271981 p T^{9} + 10147444 p^{2} T^{10} + 883587 p^{3} T^{11} + 73844 p^{4} T^{12} + 5517 p^{5} T^{13} + 364 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.74687113923873340346084095456, −2.65250326030319616812519497575, −2.53920754529625145235605714417, −2.51385970633407331468132211132, −2.42357080525057391300289368080, −2.40301638538995760014340312536, −2.34766319110349515709494804142, −2.30718987049902350782229881055, −2.10344248839087253163159067300, −2.09654277553675882851201159503, −1.93777852291883203943847208865, −1.91455134089911336377150820316, −1.77598788969883523576702958476, −1.58379092683059452437102480819, −1.36313181518439232958754855129, −1.34136392755590318876423070425, −1.30485675996269626144212694451, −1.28899110641147382908189586986, −1.16680120506906394764470465326, −1.14341872044269555472647384517, −0.998358224457116856124306351672, −0.981964054686185985015671581216, −0.889148165910405021091991329025, −0.869531104122168023929601236728, −0.19219244230822939521146451841, 0.19219244230822939521146451841, 0.869531104122168023929601236728, 0.889148165910405021091991329025, 0.981964054686185985015671581216, 0.998358224457116856124306351672, 1.14341872044269555472647384517, 1.16680120506906394764470465326, 1.28899110641147382908189586986, 1.30485675996269626144212694451, 1.34136392755590318876423070425, 1.36313181518439232958754855129, 1.58379092683059452437102480819, 1.77598788969883523576702958476, 1.91455134089911336377150820316, 1.93777852291883203943847208865, 2.09654277553675882851201159503, 2.10344248839087253163159067300, 2.30718987049902350782229881055, 2.34766319110349515709494804142, 2.40301638538995760014340312536, 2.42357080525057391300289368080, 2.51385970633407331468132211132, 2.53920754529625145235605714417, 2.65250326030319616812519497575, 2.74687113923873340346084095456
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.