Dirichlet series
| L(s) = 1 | + 4·4-s − 4·7-s − 24·11-s + 6·16-s − 24·23-s + 16·25-s − 16·28-s + 48·29-s + 16·37-s − 32·43-s − 96·44-s + 16·49-s + 8·67-s + 96·77-s − 16·79-s − 96·92-s + 64·100-s + 16·109-s − 24·112-s + 192·116-s + 236·121-s + 127-s + 131-s + 137-s + 139-s + 64·148-s + 149-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1.51·7-s − 7.23·11-s + 3/2·16-s − 5.00·23-s + 16/5·25-s − 3.02·28-s + 8.91·29-s + 2.63·37-s − 4.87·43-s − 14.4·44-s + 16/7·49-s + 0.977·67-s + 10.9·77-s − 1.80·79-s − 10.0·92-s + 32/5·100-s + 1.53·109-s − 2.26·112-s + 17.8·116-s + 21.4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.26·148-s + 0.0819·149-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{64} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{64} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{64} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.04287\times 10^{15}\) |
| Root analytic conductor: | \(3.00915\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{64} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.006334215143\) |
| \(L(\frac12)\) | \(\approx\) | \(0.006334215143\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 + 4 T - 40 T^{3} - 10 p T^{4} + 36 p T^{5} + 928 T^{6} - 1004 T^{7} - 8973 T^{8} - 1004 p T^{9} + 928 p^{2} T^{10} + 36 p^{4} T^{11} - 10 p^{5} T^{12} - 40 p^{5} T^{13} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 5 | \( 1 - 16 T^{2} + 114 T^{4} - 512 T^{6} + 1409 T^{8} + 528 p T^{10} - 2558 p^{2} T^{12} + 3712 p^{3} T^{14} - 3996 p^{4} T^{16} + 3712 p^{5} T^{18} - 2558 p^{6} T^{20} + 528 p^{7} T^{22} + 1409 p^{8} T^{24} - 512 p^{10} T^{26} + 114 p^{12} T^{28} - 16 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 + 12 T + 98 T^{2} + 600 T^{3} + 3100 T^{4} + 14040 T^{5} + 57260 T^{6} + 213828 T^{7} + 735883 T^{8} + 213828 p T^{9} + 57260 p^{2} T^{10} + 14040 p^{3} T^{11} + 3100 p^{4} T^{12} + 600 p^{5} T^{13} + 98 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( 1 + 20 T^{2} - 282 T^{4} - 5912 T^{6} + 569 p^{2} T^{8} + 1381656 T^{10} - 16803626 T^{12} - 5525956 p T^{14} + 3692424420 T^{16} - 5525956 p^{3} T^{18} - 16803626 p^{4} T^{20} + 1381656 p^{6} T^{22} + 569 p^{10} T^{24} - 5912 p^{10} T^{26} - 282 p^{12} T^{28} + 20 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 76 T^{2} + 10 p^{2} T^{4} + 74032 T^{6} + 1426627 T^{8} + 74032 p^{2} T^{10} + 10 p^{6} T^{12} + 76 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( ( 1 + 4 T^{2} - 44 T^{4} - 236 T^{6} + 201322 T^{8} - 236 p^{2} T^{10} - 44 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 + 12 T + 146 T^{2} + 1176 T^{3} + 9340 T^{4} + 59400 T^{5} + 368876 T^{6} + 1950036 T^{7} + 10059259 T^{8} + 1950036 p T^{9} + 368876 p^{2} T^{10} + 59400 p^{3} T^{11} + 9340 p^{4} T^{12} + 1176 p^{5} T^{13} + 146 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 24 T + 350 T^{2} - 3792 T^{3} + 33817 T^{4} - 260064 T^{5} + 1770734 T^{6} - 10891992 T^{7} + 61146052 T^{8} - 10891992 p T^{9} + 1770734 p^{2} T^{10} - 260064 p^{3} T^{11} + 33817 p^{4} T^{12} - 3792 p^{5} T^{13} + 350 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( 1 + 68 T^{2} + 2124 T^{4} - 13208 T^{6} - 3130954 T^{8} - 3600468 p T^{10} - 856945376 T^{12} + 65916599036 T^{14} + 3453064500195 T^{16} + 65916599036 p^{2} T^{18} - 856945376 p^{4} T^{20} - 3600468 p^{7} T^{22} - 3130954 p^{8} T^{24} - 13208 p^{10} T^{26} + 2124 p^{12} T^{28} + 68 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 - 4 T + 64 T^{2} - 16 p T^{3} + 2071 T^{4} - 16 p^{2} T^{5} + 64 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 41 | \( 1 - 88 T^{2} - 516 T^{4} + 113968 T^{6} + 9403466 T^{8} - 337914312 T^{10} - 19044119696 T^{12} + 26706789464 T^{14} + 52072549508115 T^{16} + 26706789464 p^{2} T^{18} - 19044119696 p^{4} T^{20} - 337914312 p^{6} T^{22} + 9403466 p^{8} T^{24} + 113968 p^{10} T^{26} - 516 p^{12} T^{28} - 88 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 + 16 T + 60 T^{2} + 224 T^{3} + 4298 T^{4} + 1680 T^{5} - 278096 T^{6} - 1714928 T^{7} - 7783965 T^{8} - 1714928 p T^{9} - 278096 p^{2} T^{10} + 1680 p^{3} T^{11} + 4298 p^{4} T^{12} + 224 p^{5} T^{13} + 60 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 28 T^{2} - 60 p T^{4} + 442792 T^{6} - 785482 T^{8} - 1235178924 T^{10} + 70052219680 T^{12} + 1859047277276 T^{14} - 195567917648157 T^{16} + 1859047277276 p^{2} T^{18} + 70052219680 p^{4} T^{20} - 1235178924 p^{6} T^{22} - 785482 p^{8} T^{24} + 442792 p^{10} T^{26} - 60 p^{13} T^{28} - 28 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 140 T^{2} + 9546 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{4} \) | |
| 59 | \( 1 - 412 T^{2} + 93084 T^{4} - 14645528 T^{6} + 1776790166 T^{8} - 175548730668 T^{10} + 14625889667104 T^{12} - 1051794756826564 T^{14} + 66181103958774915 T^{16} - 1051794756826564 p^{2} T^{18} + 14625889667104 p^{4} T^{20} - 175548730668 p^{6} T^{22} + 1776790166 p^{8} T^{24} - 14645528 p^{10} T^{26} + 93084 p^{12} T^{28} - 412 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 + 140 T^{2} + 8742 T^{4} + 377944 T^{6} + 12716321 T^{8} + 1262185512 T^{10} + 55394759446 T^{12} - 3479572730284 T^{14} - 381624798015900 T^{16} - 3479572730284 p^{2} T^{18} + 55394759446 p^{4} T^{20} + 1262185512 p^{6} T^{22} + 12716321 p^{8} T^{24} + 377944 p^{10} T^{26} + 8742 p^{12} T^{28} + 140 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 4 T - 150 T^{2} - 200 T^{3} + 13604 T^{4} + 50136 T^{5} - 679988 T^{6} - 2044708 T^{7} + 31792779 T^{8} - 2044708 p T^{9} - 679988 p^{2} T^{10} + 50136 p^{3} T^{11} + 13604 p^{4} T^{12} - 200 p^{5} T^{13} - 150 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 208 T^{2} + 20044 T^{4} - 919600 T^{6} + 38018662 T^{8} - 919600 p^{2} T^{10} + 20044 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 416 T^{2} + 85006 T^{4} - 10920368 T^{6} + 955245667 T^{8} - 10920368 p^{2} T^{10} + 85006 p^{4} T^{12} - 416 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 8 T - 186 T^{2} - 1784 T^{3} + 20204 T^{4} + 187908 T^{5} - 1285580 T^{6} - 6920968 T^{7} + 88968555 T^{8} - 6920968 p T^{9} - 1285580 p^{2} T^{10} + 187908 p^{3} T^{11} + 20204 p^{4} T^{12} - 1784 p^{5} T^{13} - 186 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 244 T^{2} + 29292 T^{4} - 1892744 T^{6} + 14025494 T^{8} + 10996760316 T^{10} - 1209533879264 T^{12} + 78676809080852 T^{14} - 5088148403639229 T^{16} + 78676809080852 p^{2} T^{18} - 1209533879264 p^{4} T^{20} + 10996760316 p^{6} T^{22} + 14025494 p^{8} T^{24} - 1892744 p^{10} T^{26} + 29292 p^{12} T^{28} - 244 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 172 T^{2} + 12970 T^{4} + 1141936 T^{6} + 124459747 T^{8} + 1141936 p^{2} T^{10} + 12970 p^{4} T^{12} + 172 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 656 T^{2} + 235356 T^{4} + 58506976 T^{6} + 11171002826 T^{8} + 1735354359984 T^{10} + 228321888182896 T^{12} + 26232085033006448 T^{14} + 2685317685162684435 T^{16} + 26232085033006448 p^{2} T^{18} + 228321888182896 p^{4} T^{20} + 1735354359984 p^{6} T^{22} + 11171002826 p^{8} T^{24} + 58506976 p^{10} T^{26} + 235356 p^{12} T^{28} + 656 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.51621632194150989288260724008, −2.45832167051646558896181733948, −2.45688692102470240597407044959, −2.45672185258167667222358226654, −2.40947099092654019930607663200, −2.37940504812253487284901177266, −2.10582790067469326032395706012, −2.00420360242172292481985240907, −1.98679248653204189475467732891, −1.96193421564314812561694922365, −1.87508498626041036779905217503, −1.73175251567114195800861364484, −1.54489030971766222611493116433, −1.26640950433230278488988365591, −1.22963104329373173553171848367, −1.17645882069965356281134362815, −1.17240062395425003230860690903, −1.02817214272859019795695638709, −1.02354492120709224323734948869, −0.986129821918159246659957371539, −0.77461231996064996865246980769, −0.32531403626535454791224959385, −0.26699796638194683192415849189, −0.094195536625100030746116560419, −0.02765513535330380587215478961, 0.02765513535330380587215478961, 0.094195536625100030746116560419, 0.26699796638194683192415849189, 0.32531403626535454791224959385, 0.77461231996064996865246980769, 0.986129821918159246659957371539, 1.02354492120709224323734948869, 1.02817214272859019795695638709, 1.17240062395425003230860690903, 1.17645882069965356281134362815, 1.22963104329373173553171848367, 1.26640950433230278488988365591, 1.54489030971766222611493116433, 1.73175251567114195800861364484, 1.87508498626041036779905217503, 1.96193421564314812561694922365, 1.98679248653204189475467732891, 2.00420360242172292481985240907, 2.10582790067469326032395706012, 2.37940504812253487284901177266, 2.40947099092654019930607663200, 2.45672185258167667222358226654, 2.45688692102470240597407044959, 2.45832167051646558896181733948, 2.51621632194150989288260724008
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.