Dirichlet series
| L(s) = 1 | − 6·4-s − 2·5-s − 5·9-s + 25·16-s − 6·19-s + 12·20-s − 6·25-s − 2·29-s + 8·31-s + 30·36-s + 52·41-s + 10·45-s − 19·49-s + 2·59-s + 40·61-s − 74·64-s + 36·71-s + 36·76-s − 38·79-s − 50·80-s + 31·81-s + 24·89-s + 12·95-s + 36·100-s + 94·101-s − 20·109-s + 12·116-s + ⋯ |
| L(s) = 1 | − 3·4-s − 0.894·5-s − 5/3·9-s + 25/4·16-s − 1.37·19-s + 2.68·20-s − 6/5·25-s − 0.371·29-s + 1.43·31-s + 5·36-s + 8.12·41-s + 1.49·45-s − 2.71·49-s + 0.260·59-s + 5.12·61-s − 9.25·64-s + 4.27·71-s + 4.12·76-s − 4.27·79-s − 5.59·80-s + 31/9·81-s + 2.54·89-s + 1.23·95-s + 18/5·100-s + 9.35·101-s − 1.91·109-s + 1.11·116-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(5^{16} \cdot 11^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.80075\times 10^{10}\) |
| Root analytic conductor: | \(2.19794\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 5^{16} \cdot 11^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.906728102\) |
| \(L(\frac12)\) | \(\approx\) | \(1.906728102\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 + 2 T + 2 p T^{2} + 42 T^{3} + 99 T^{4} + 338 T^{5} + 36 p^{2} T^{6} + 2008 T^{7} + 5281 T^{8} + 2008 p T^{9} + 36 p^{4} T^{10} + 338 p^{3} T^{11} + 99 p^{4} T^{12} + 42 p^{5} T^{13} + 2 p^{7} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + 3 p T^{2} + 11 T^{4} - 5 p T^{6} - 77 T^{8} - 55 p T^{10} + 69 T^{12} + 129 p^{2} T^{14} + 1193 T^{16} + 129 p^{4} T^{18} + 69 p^{4} T^{20} - 55 p^{7} T^{22} - 77 p^{8} T^{24} - 5 p^{11} T^{26} + 11 p^{12} T^{28} + 3 p^{15} T^{30} + p^{16} T^{32} \) |
| 3 | \( 1 + 5 T^{2} - 2 p T^{4} - 140 T^{6} - 10 p^{3} T^{8} + 1165 T^{10} + 5506 T^{12} - 3770 T^{14} - 53741 T^{16} - 3770 p^{2} T^{18} + 5506 p^{4} T^{20} + 1165 p^{6} T^{22} - 10 p^{11} T^{24} - 140 p^{10} T^{26} - 2 p^{13} T^{28} + 5 p^{14} T^{30} + p^{16} T^{32} \) | |
| 7 | \( 1 + 19 T^{2} + 229 T^{4} + 2109 T^{6} + 2213 p T^{8} + 92002 T^{10} + 438866 T^{12} + 1927012 T^{14} + 12254167 T^{16} + 1927012 p^{2} T^{18} + 438866 p^{4} T^{20} + 92002 p^{6} T^{22} + 2213 p^{9} T^{24} + 2109 p^{10} T^{26} + 229 p^{12} T^{28} + 19 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( 1 + 42 T^{2} + 635 T^{4} - 275 T^{6} - 168035 T^{8} - 2770249 T^{10} - 11553003 T^{12} + 300462090 T^{14} + 6477797525 T^{16} + 300462090 p^{2} T^{18} - 11553003 p^{4} T^{20} - 2770249 p^{6} T^{22} - 168035 p^{8} T^{24} - 275 p^{10} T^{26} + 635 p^{12} T^{28} + 42 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 5 T^{2} - 186 T^{4} - 10315 T^{6} + 52680 T^{8} + 4287860 T^{10} + 38701866 T^{12} - 14826280 p T^{14} - 31527815401 T^{16} - 14826280 p^{3} T^{18} + 38701866 p^{4} T^{20} + 4287860 p^{6} T^{22} + 52680 p^{8} T^{24} - 10315 p^{10} T^{26} - 186 p^{12} T^{28} - 5 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 3 T - 35 T^{2} - 180 T^{3} + 185 T^{4} + 4194 T^{5} + 14942 T^{6} - 37935 T^{7} - 448355 T^{8} - 37935 p T^{9} + 14942 p^{2} T^{10} + 4194 p^{3} T^{11} + 185 p^{4} T^{12} - 180 p^{5} T^{13} - 35 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 85 T^{2} + 3382 T^{4} - 84765 T^{6} + 1856153 T^{8} - 84765 p^{2} T^{10} + 3382 p^{4} T^{12} - 85 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 + T - 33 T^{2} - 278 T^{3} + 1127 T^{4} - 610 T^{5} - 18852 T^{6} - 40501 T^{7} + 2275055 T^{8} - 40501 p T^{9} - 18852 p^{2} T^{10} - 610 p^{3} T^{11} + 1127 p^{4} T^{12} - 278 p^{5} T^{13} - 33 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 4 T - 23 T^{2} - 101 T^{3} + 2551 T^{4} - 4925 T^{5} - 57287 T^{6} - 61630 T^{7} + 3347567 T^{8} - 61630 p T^{9} - 57287 p^{2} T^{10} - 4925 p^{3} T^{11} + 2551 p^{4} T^{12} - 101 p^{5} T^{13} - 23 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 120 T^{2} + 6419 T^{4} + 169710 T^{6} + 1135850 T^{8} - 4332930 p T^{10} - 10706145719 T^{12} - 404428360410 T^{14} - 12797968713881 T^{16} - 404428360410 p^{2} T^{18} - 10706145719 p^{4} T^{20} - 4332930 p^{7} T^{22} + 1135850 p^{8} T^{24} + 169710 p^{10} T^{26} + 6419 p^{12} T^{28} + 120 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 26 T + 263 T^{2} - 1156 T^{3} - 1228 T^{4} + 54770 T^{5} - 8553 p T^{6} + 226372 T^{7} + 7159995 T^{8} + 226372 p T^{9} - 8553 p^{3} T^{10} + 54770 p^{3} T^{11} - 1228 p^{4} T^{12} - 1156 p^{5} T^{13} + 263 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 171 T^{2} + 15457 T^{4} - 1005398 T^{6} + 49870565 T^{8} - 1005398 p^{2} T^{10} + 15457 p^{4} T^{12} - 171 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 210 T^{2} + 23017 T^{4} + 1971945 T^{6} + 143287783 T^{8} + 9109735985 T^{10} + 538571627359 T^{12} + 29130862463950 T^{14} + 1425554876818805 T^{16} + 29130862463950 p^{2} T^{18} + 538571627359 p^{4} T^{20} + 9109735985 p^{6} T^{22} + 143287783 p^{8} T^{24} + 1971945 p^{10} T^{26} + 23017 p^{12} T^{28} + 210 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 5 T^{2} - 981 T^{4} - 139345 T^{6} - 1908885 T^{8} + 84123170 T^{10} + 22219415316 T^{12} - 356460233840 T^{14} - 33514746895621 T^{16} - 356460233840 p^{2} T^{18} + 22219415316 p^{4} T^{20} + 84123170 p^{6} T^{22} - 1908885 p^{8} T^{24} - 139345 p^{10} T^{26} - 981 p^{12} T^{28} - 5 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 - T + 11 T^{2} + 85 T^{3} + 7113 T^{4} + 12500 T^{5} - 215226 T^{6} + 1526554 T^{7} + 21070553 T^{8} + 1526554 p T^{9} - 215226 p^{2} T^{10} + 12500 p^{3} T^{11} + 7113 p^{4} T^{12} + 85 p^{5} T^{13} + 11 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 20 T + 26 T^{2} + 2140 T^{3} - 12685 T^{4} - 139300 T^{5} + 1704084 T^{6} + 3518720 T^{7} - 128652411 T^{8} + 3518720 p T^{9} + 1704084 p^{2} T^{10} - 139300 p^{3} T^{11} - 12685 p^{4} T^{12} + 2140 p^{5} T^{13} + 26 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 387 T^{2} + 71530 T^{4} - 8300957 T^{6} + 662638709 T^{8} - 8300957 p^{2} T^{10} + 71530 p^{4} T^{12} - 387 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 18 T + 119 T^{2} - 660 T^{3} + 6528 T^{4} - 48510 T^{5} + 528511 T^{6} - 8889912 T^{7} + 95792243 T^{8} - 8889912 p T^{9} + 528511 p^{2} T^{10} - 48510 p^{3} T^{11} + 6528 p^{4} T^{12} - 660 p^{5} T^{13} + 119 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 91 T^{2} - 1406 T^{4} - 1338589 T^{6} - 44180104 T^{8} + 6397599698 T^{10} + 733001427326 T^{12} - 18973235763482 T^{14} - 4254917733021293 T^{16} - 18973235763482 p^{2} T^{18} + 733001427326 p^{4} T^{20} + 6397599698 p^{6} T^{22} - 44180104 p^{8} T^{24} - 1338589 p^{10} T^{26} - 1406 p^{12} T^{28} + 91 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 19 T + 52 T^{2} - 772 T^{3} + 1222 T^{4} + 44105 T^{5} - 607892 T^{6} - 7482254 T^{7} - 37383265 T^{8} - 7482254 p T^{9} - 607892 p^{2} T^{10} + 44105 p^{3} T^{11} + 1222 p^{4} T^{12} - 772 p^{5} T^{13} + 52 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 + 299 T^{2} + 47629 T^{4} + 5098189 T^{6} + 383737301 T^{8} + 24754599712 T^{10} + 1861275246176 T^{12} + 181162480129562 T^{14} + 17063861270507407 T^{16} + 181162480129562 p^{2} T^{18} + 1861275246176 p^{4} T^{20} + 24754599712 p^{6} T^{22} + 383737301 p^{8} T^{24} + 5098189 p^{10} T^{26} + 47629 p^{12} T^{28} + 299 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 6 T + 228 T^{2} - 1116 T^{3} + 26613 T^{4} - 1116 p T^{5} + 228 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 97 | \( 1 + 412 T^{2} + 78833 T^{4} + 8589772 T^{6} + 459983126 T^{8} - 15760400260 T^{10} - 6348990508023 T^{12} - 840119161411750 T^{14} - 86547769155085413 T^{16} - 840119161411750 p^{2} T^{18} - 6348990508023 p^{4} T^{20} - 15760400260 p^{6} T^{22} + 459983126 p^{8} T^{24} + 8589772 p^{10} T^{26} + 78833 p^{12} T^{28} + 412 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.82281347127344484814846581576, −2.72106619707846373852440846859, −2.67672901939861978028320101472, −2.64904693899437268996673817454, −2.57967521742358309286089004646, −2.49565299517664100862310801815, −2.46793321726129781113476644827, −2.35059051770324192165269264817, −2.19721656919579392852090750660, −2.08212313841406821085477471350, −2.07652539280701263387904681000, −1.88013223124651688201112591216, −1.85538983489232495567484363773, −1.62516872156712848680296527896, −1.56929081696194904865278608623, −1.36146362287046161308518092794, −1.30136988302279891189027460927, −1.17554990902077992120733454787, −1.00747074389092170506248316985, −0.75011902987161056402503347395, −0.66347527096791007356243182905, −0.65524425543414335256715691785, −0.61298421803716422509177679568, −0.49629664067860801616809504780, −0.17967563185583109491568685734, 0.17967563185583109491568685734, 0.49629664067860801616809504780, 0.61298421803716422509177679568, 0.65524425543414335256715691785, 0.66347527096791007356243182905, 0.75011902987161056402503347395, 1.00747074389092170506248316985, 1.17554990902077992120733454787, 1.30136988302279891189027460927, 1.36146362287046161308518092794, 1.56929081696194904865278608623, 1.62516872156712848680296527896, 1.85538983489232495567484363773, 1.88013223124651688201112591216, 2.07652539280701263387904681000, 2.08212313841406821085477471350, 2.19721656919579392852090750660, 2.35059051770324192165269264817, 2.46793321726129781113476644827, 2.49565299517664100862310801815, 2.57967521742358309286089004646, 2.64904693899437268996673817454, 2.67672901939861978028320101472, 2.72106619707846373852440846859, 2.82281347127344484814846581576
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.