Dirichlet series
| L(s) = 1 | − 10·4-s − 12·9-s + 40·11-s + 61·16-s + 200·29-s − 252·31-s + 120·36-s − 400·44-s − 38·49-s + 108·59-s − 792·61-s − 270·64-s + 328·71-s + 412·79-s + 54·81-s − 564·89-s − 480·99-s − 804·101-s + 124·109-s − 2.00e3·116-s + 1.49e3·121-s + 2.52e3·124-s + 127-s + 131-s + 137-s + 139-s − 732·144-s + ⋯ |
| L(s) = 1 | − 5/2·4-s − 4/3·9-s + 3.63·11-s + 3.81·16-s + 6.89·29-s − 8.12·31-s + 10/3·36-s − 9.09·44-s − 0.775·49-s + 1.83·59-s − 12.9·61-s − 4.21·64-s + 4.61·71-s + 5.21·79-s + 2/3·81-s − 6.33·89-s − 4.84·99-s − 7.96·101-s + 1.13·109-s − 17.2·116-s + 12.3·121-s + 20.3·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 5.08·144-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 5^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.07545\times 10^{18}\) |
| Root analytic conductor: | \(3.78222\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 5^{32} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(3.701863879\) |
| \(L(\frac12)\) | \(\approx\) | \(3.701863879\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 5 | \( 1 \) | |
| 7 | \( 1 + 38 T^{2} + 121 T^{4} - 2698 p^{2} T^{6} - 3116 p^{4} T^{8} - 2698 p^{6} T^{10} + 121 p^{8} T^{12} + 38 p^{12} T^{14} + p^{16} T^{16} \) | |
| good | 2 | \( 1 + 5 p T^{2} + 39 T^{4} + 25 p T^{6} - 235 T^{8} - 195 p^{3} T^{10} - 1421 p^{2} T^{12} - 805 p^{5} T^{14} - 7311 p^{4} T^{16} - 805 p^{9} T^{18} - 1421 p^{10} T^{20} - 195 p^{15} T^{22} - 235 p^{16} T^{24} + 25 p^{21} T^{26} + 39 p^{24} T^{28} + 5 p^{29} T^{30} + p^{32} T^{32} \) |
| 11 | \( ( 1 - 20 T - 147 T^{2} + 2960 T^{3} + 57131 T^{4} - 519480 T^{5} - 8882912 T^{6} + 8003440 T^{7} + 1602642534 T^{8} + 8003440 p^{2} T^{9} - 8882912 p^{4} T^{10} - 519480 p^{6} T^{11} + 57131 p^{8} T^{12} + 2960 p^{10} T^{13} - 147 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 188 T^{2} + 39826 T^{4} + 8798048 T^{6} + 2113175419 T^{8} + 8798048 p^{4} T^{10} + 39826 p^{8} T^{12} + 188 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 - 1208 T^{2} + 685248 T^{4} - 238401232 T^{6} + 61817983778 T^{8} - 17147293777272 T^{10} + 6777126590126848 T^{12} - 2780839422001632392 T^{14} + \)\(91\!\cdots\!83\)\( T^{16} - 2780839422001632392 p^{4} T^{18} + 6777126590126848 p^{8} T^{20} - 17147293777272 p^{12} T^{22} + 61817983778 p^{16} T^{24} - 238401232 p^{20} T^{26} + 685248 p^{24} T^{28} - 1208 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( ( 1 + 598 T^{2} + 183481 T^{4} + 682560 T^{5} - 48973562 T^{6} + 501474240 T^{7} - 32269961996 T^{8} + 501474240 p^{2} T^{9} - 48973562 p^{4} T^{10} + 682560 p^{6} T^{11} + 183481 p^{8} T^{12} + 598 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 + 850 T^{2} + 592959 T^{4} + 52476230 T^{6} - 67877437675 T^{8} - 90481750443780 T^{10} + 6471161742853546 T^{12} + 26894707124495548840 T^{14} + \)\(27\!\cdots\!14\)\( T^{16} + 26894707124495548840 p^{4} T^{18} + 6471161742853546 p^{8} T^{20} - 90481750443780 p^{12} T^{22} - 67877437675 p^{16} T^{24} + 52476230 p^{20} T^{26} + 592959 p^{24} T^{28} + 850 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( ( 1 - 50 T + 1234 T^{2} + 15850 T^{3} - 1164374 T^{4} + 15850 p^{2} T^{5} + 1234 p^{4} T^{6} - 50 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 31 | \( ( 1 + 126 T + 9883 T^{2} + 578466 T^{3} + 27206317 T^{4} + 1079090100 T^{5} + 38160094402 T^{6} + 1243487527488 T^{7} + 38998740329170 T^{8} + 1243487527488 p^{2} T^{9} + 38160094402 p^{4} T^{10} + 1079090100 p^{6} T^{11} + 27206317 p^{8} T^{12} + 578466 p^{10} T^{13} + 9883 p^{12} T^{14} + 126 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 37 | \( 1 + 4012 T^{2} + 3715446 T^{4} + 1106164568 T^{6} + 16621668655361 T^{8} + 31297439550751656 T^{10} + 8237655295148593894 T^{12} + \)\(26\!\cdots\!24\)\( T^{14} + \)\(95\!\cdots\!52\)\( T^{16} + \)\(26\!\cdots\!24\)\( p^{4} T^{18} + 8237655295148593894 p^{8} T^{20} + 31297439550751656 p^{12} T^{22} + 16621668655361 p^{16} T^{24} + 1106164568 p^{20} T^{26} + 3715446 p^{24} T^{28} + 4012 p^{28} T^{30} + p^{32} T^{32} \) | |
| 41 | \( ( 1 - 10106 T^{2} + 48877645 T^{4} - 146585251874 T^{6} + 296639674915264 T^{8} - 146585251874 p^{4} T^{10} + 48877645 p^{8} T^{12} - 10106 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 3058 T^{2} + 2023873 T^{4} - 9897938386 T^{6} + 38630411096740 T^{8} - 9897938386 p^{4} T^{10} + 2023873 p^{8} T^{12} - 3058 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 47 | \( 1 - 16718 T^{2} + 155339283 T^{4} - 1005045086002 T^{6} + 4999251300672953 T^{8} - 20050901794507340652 T^{10} + \)\(66\!\cdots\!18\)\( T^{12} - \)\(18\!\cdots\!72\)\( T^{14} + \)\(44\!\cdots\!78\)\( T^{16} - \)\(18\!\cdots\!72\)\( p^{4} T^{18} + \)\(66\!\cdots\!18\)\( p^{8} T^{20} - 20050901794507340652 p^{12} T^{22} + 4999251300672953 p^{16} T^{24} - 1005045086002 p^{20} T^{26} + 155339283 p^{24} T^{28} - 16718 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 + 11914 T^{2} + 81330459 T^{4} + 353520432038 T^{6} + 995406028387193 T^{8} + 1125584826270060372 T^{10} - \)\(50\!\cdots\!38\)\( T^{12} - \)\(36\!\cdots\!48\)\( T^{14} - \)\(12\!\cdots\!02\)\( T^{16} - \)\(36\!\cdots\!48\)\( p^{4} T^{18} - \)\(50\!\cdots\!38\)\( p^{8} T^{20} + 1125584826270060372 p^{12} T^{22} + 995406028387193 p^{16} T^{24} + 353520432038 p^{20} T^{26} + 81330459 p^{24} T^{28} + 11914 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( ( 1 - 54 T + 122 p T^{2} - 336204 T^{3} + 19932742 T^{4} + 202333950 T^{5} - 35478676088 T^{6} + 7641841019598 T^{7} - 385856896323245 T^{8} + 7641841019598 p^{2} T^{9} - 35478676088 p^{4} T^{10} + 202333950 p^{6} T^{11} + 19932742 p^{8} T^{12} - 336204 p^{10} T^{13} + 122 p^{13} T^{14} - 54 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 61 | \( ( 1 + 396 T + 83164 T^{2} + 12233232 T^{3} + 1413738778 T^{4} + 136250283708 T^{5} + 11318984386192 T^{6} + 825650586150588 T^{7} + 53403008176121923 T^{8} + 825650586150588 p^{2} T^{9} + 11318984386192 p^{4} T^{10} + 136250283708 p^{6} T^{11} + 1413738778 p^{8} T^{12} + 12233232 p^{10} T^{13} + 83164 p^{12} T^{14} + 396 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 67 | \( 1 + 21394 T^{2} + 239151459 T^{4} + 1707944285678 T^{6} + 8446046232738713 T^{8} + 30181218213405933492 T^{10} + \)\(84\!\cdots\!02\)\( T^{12} + \)\(24\!\cdots\!52\)\( T^{14} + \)\(90\!\cdots\!38\)\( T^{16} + \)\(24\!\cdots\!52\)\( p^{4} T^{18} + \)\(84\!\cdots\!02\)\( p^{8} T^{20} + 30181218213405933492 p^{12} T^{22} + 8446046232738713 p^{16} T^{24} + 1707944285678 p^{20} T^{26} + 239151459 p^{24} T^{28} + 21394 p^{28} T^{30} + p^{32} T^{32} \) | |
| 71 | \( ( 1 - 82 T + 12166 T^{2} - 846262 T^{3} + 94594474 T^{4} - 846262 p^{2} T^{5} + 12166 p^{4} T^{6} - 82 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 73 | \( 1 - 15422 T^{2} + 48442059 T^{4} - 65292445378 T^{6} + 4729429873870505 T^{8} - 29601647342638120044 T^{10} - \)\(89\!\cdots\!26\)\( T^{12} - \)\(37\!\cdots\!20\)\( T^{14} + \)\(66\!\cdots\!90\)\( T^{16} - \)\(37\!\cdots\!20\)\( p^{4} T^{18} - \)\(89\!\cdots\!26\)\( p^{8} T^{20} - 29601647342638120044 p^{12} T^{22} + 4729429873870505 p^{16} T^{24} - 65292445378 p^{20} T^{26} + 48442059 p^{24} T^{28} - 15422 p^{28} T^{30} + p^{32} T^{32} \) | |
| 79 | \( ( 1 - 206 T + 5583 T^{2} + 659438 T^{3} + 124066817 T^{4} - 19494076044 T^{5} + 428008398310 T^{6} - 33090623674568 T^{7} + 7605703397631354 T^{8} - 33090623674568 p^{2} T^{9} + 428008398310 p^{4} T^{10} - 19494076044 p^{6} T^{11} + 124066817 p^{8} T^{12} + 659438 p^{10} T^{13} + 5583 p^{12} T^{14} - 206 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 20672 T^{2} + 223804480 T^{4} + 2182268545136 T^{6} + 17948924233578718 T^{8} + 2182268545136 p^{4} T^{10} + 223804480 p^{8} T^{12} + 20672 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 282 T + 59686 T^{2} + 9356196 T^{3} + 1240796086 T^{4} + 138656838366 T^{5} + 14271061565800 T^{6} + 1337157406377822 T^{7} + 121622616146107507 T^{8} + 1337157406377822 p^{2} T^{9} + 14271061565800 p^{4} T^{10} + 138656838366 p^{6} T^{11} + 1240796086 p^{8} T^{12} + 9356196 p^{10} T^{13} + 59686 p^{12} T^{14} + 282 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 44576 T^{2} + 925514428 T^{4} + 12414040936928 T^{6} + 128325632901816454 T^{8} + 12414040936928 p^{4} T^{10} + 925514428 p^{8} T^{12} + 44576 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
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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
−2.80905485727286001169291721413, −2.58744330396297700953304484832, −2.57199005533716220240484468818, −2.33335009224546788160076166230, −2.29563165199719240726526311111, −2.24041079322249840432021976166, −2.17199033186667841161617861022, −1.91184539794260964585272647329, −1.84628683128124908410439755772, −1.74618199965417473330286896237, −1.69542338258220275513054044487, −1.58310383079724547446375982058, −1.56198481222479898834227497044, −1.44549830161778842475983446120, −1.38916138298356743725912209225, −1.37656166728283188402799753012, −1.00873562666584508818126410010, −0.996567947887048197212949720418, −0.826210593367603629790710791911, −0.73076028479251251055118035853, −0.71492932881541901133478998279, −0.35149990150719512838581982176, −0.31680543685367042291021819925, −0.30284940152413879887315589686, −0.14196050178427725761691696233, 0.14196050178427725761691696233, 0.30284940152413879887315589686, 0.31680543685367042291021819925, 0.35149990150719512838581982176, 0.71492932881541901133478998279, 0.73076028479251251055118035853, 0.826210593367603629790710791911, 0.996567947887048197212949720418, 1.00873562666584508818126410010, 1.37656166728283188402799753012, 1.38916138298356743725912209225, 1.44549830161778842475983446120, 1.56198481222479898834227497044, 1.58310383079724547446375982058, 1.69542338258220275513054044487, 1.74618199965417473330286896237, 1.84628683128124908410439755772, 1.91184539794260964585272647329, 2.17199033186667841161617861022, 2.24041079322249840432021976166, 2.29563165199719240726526311111, 2.33335009224546788160076166230, 2.57199005533716220240484468818, 2.58744330396297700953304484832, 2.80905485727286001169291721413
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.