Dirichlet series
| L(s) = 1 | + 13·4-s − 14·5-s − 45·9-s + 132·11-s + 54·16-s − 348·19-s − 182·20-s − 89·25-s − 740·29-s + 684·31-s − 585·36-s + 1.60e3·41-s + 1.71e3·44-s + 630·45-s − 245·49-s − 1.84e3·55-s − 1.40e3·59-s + 1.30e3·61-s + 358·64-s + 2.94e3·71-s − 4.52e3·76-s + 1.64e3·79-s − 756·80-s + 1.21e3·81-s − 572·89-s + 4.87e3·95-s − 5.94e3·99-s + ⋯ |
| L(s) = 1 | + 13/8·4-s − 1.25·5-s − 5/3·9-s + 3.61·11-s + 0.843·16-s − 4.20·19-s − 2.03·20-s − 0.711·25-s − 4.73·29-s + 3.96·31-s − 2.70·36-s + 6.10·41-s + 5.87·44-s + 2.08·45-s − 5/7·49-s − 4.53·55-s − 3.10·59-s + 2.72·61-s + 0.699·64-s + 4.91·71-s − 6.82·76-s + 2.33·79-s − 1.05·80-s + 5/3·81-s − 0.681·89-s + 5.26·95-s − 6.03·99-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 5^{10} \cdot 7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.32824\times 10^{7}\) |
| Root analytic conductor: | \(2.48901\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 5^{10} \cdot 7^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(1.323380540\) |
| \(L(\frac12)\) | \(\approx\) | \(1.323380540\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{2} T^{2} )^{5} \) |
| 5 | \( 1 + 14 T + 57 p T^{2} + 1152 T^{3} + 566 p T^{4} - 7388 p^{2} T^{5} + 566 p^{4} T^{6} + 1152 p^{6} T^{7} + 57 p^{10} T^{8} + 14 p^{12} T^{9} + p^{15} T^{10} \) | |
| 7 | \( ( 1 + p^{2} T^{2} )^{5} \) | |
| good | 2 | \( 1 - 13 T^{2} + 115 T^{4} - 1151 T^{6} + 187 p^{6} T^{8} - 895 p^{7} T^{10} + 187 p^{12} T^{12} - 1151 p^{12} T^{14} + 115 p^{18} T^{16} - 13 p^{24} T^{18} + p^{30} T^{20} \) |
| 11 | \( ( 1 - 6 p T + 4555 T^{2} - 210928 T^{3} + 10802758 T^{4} - 383496700 T^{5} + 10802758 p^{3} T^{6} - 210928 p^{6} T^{7} + 4555 p^{9} T^{8} - 6 p^{13} T^{9} + p^{15} T^{10} )^{2} \) | |
| 13 | \( 1 - 13662 T^{2} + 97034245 T^{4} - 34951648968 p T^{6} + 1533415610487218 T^{8} - 3873561618402720820 T^{10} + 1533415610487218 p^{6} T^{12} - 34951648968 p^{13} T^{14} + 97034245 p^{18} T^{16} - 13662 p^{24} T^{18} + p^{30} T^{20} \) | |
| 17 | \( 1 - 3586 T^{2} - 9840163 T^{4} - 226703927544 T^{6} + 917313320435074 T^{8} + 2772044023815717620 T^{10} + 917313320435074 p^{6} T^{12} - 226703927544 p^{12} T^{14} - 9840163 p^{18} T^{16} - 3586 p^{24} T^{18} + p^{30} T^{20} \) | |
| 19 | \( ( 1 + 174 T + 30655 T^{2} + 3260120 T^{3} + 355132498 T^{4} + 29134734196 T^{5} + 355132498 p^{3} T^{6} + 3260120 p^{6} T^{7} + 30655 p^{9} T^{8} + 174 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 23 | \( 1 - 2994 p T^{2} + 2443463533 T^{4} - 58952757931656 T^{6} + 1054996500313105346 T^{8} - \)\(14\!\cdots\!64\)\( T^{10} + 1054996500313105346 p^{6} T^{12} - 58952757931656 p^{12} T^{14} + 2443463533 p^{18} T^{16} - 2994 p^{25} T^{18} + p^{30} T^{20} \) | |
| 29 | \( ( 1 + 370 T + 160385 T^{2} + 37125560 T^{3} + 8745786530 T^{4} + 1370740158092 T^{5} + 8745786530 p^{3} T^{6} + 37125560 p^{6} T^{7} + 160385 p^{9} T^{8} + 370 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 31 | \( ( 1 - 342 T + 142051 T^{2} - 26927448 T^{3} + 6639779138 T^{4} - 944601593732 T^{5} + 6639779138 p^{3} T^{6} - 26927448 p^{6} T^{7} + 142051 p^{9} T^{8} - 342 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 37 | \( 1 - 160866 T^{2} + 14645741397 T^{4} - 941454249392984 T^{6} + 51314852590450172274 T^{8} - \)\(26\!\cdots\!00\)\( T^{10} + 51314852590450172274 p^{6} T^{12} - 941454249392984 p^{12} T^{14} + 14645741397 p^{18} T^{16} - 160866 p^{24} T^{18} + p^{30} T^{20} \) | |
| 41 | \( ( 1 - 802 T + 392457 T^{2} - 146273472 T^{3} + 44792512462 T^{4} - 12012058617212 T^{5} + 44792512462 p^{3} T^{6} - 146273472 p^{6} T^{7} + 392457 p^{9} T^{8} - 802 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 43 | \( 1 - 485758 T^{2} + 119488561781 T^{4} - 19372440245362728 T^{6} + \)\(22\!\cdots\!22\)\( T^{8} - \)\(20\!\cdots\!28\)\( T^{10} + \)\(22\!\cdots\!22\)\( p^{6} T^{12} - 19372440245362728 p^{12} T^{14} + 119488561781 p^{18} T^{16} - 485758 p^{24} T^{18} + p^{30} T^{20} \) | |
| 47 | \( 1 - 502182 T^{2} + 138597263629 T^{4} - 26712508734207432 T^{6} + \)\(39\!\cdots\!54\)\( T^{8} - \)\(45\!\cdots\!76\)\( T^{10} + \)\(39\!\cdots\!54\)\( p^{6} T^{12} - 26712508734207432 p^{12} T^{14} + 138597263629 p^{18} T^{16} - 502182 p^{24} T^{18} + p^{30} T^{20} \) | |
| 53 | \( 1 - 997006 T^{2} + 484698335029 T^{4} - 151727963326781416 T^{6} + \)\(34\!\cdots\!38\)\( T^{8} - \)\(57\!\cdots\!36\)\( T^{10} + \)\(34\!\cdots\!38\)\( p^{6} T^{12} - 151727963326781416 p^{12} T^{14} + 484698335029 p^{18} T^{16} - 997006 p^{24} T^{18} + p^{30} T^{20} \) | |
| 59 | \( ( 1 + 704 T + 427487 T^{2} + 197865856 T^{3} + 99740405354 T^{4} + 55609053370240 T^{5} + 99740405354 p^{3} T^{6} + 197865856 p^{6} T^{7} + 427487 p^{9} T^{8} + 704 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 61 | \( ( 1 - 650 T + 689345 T^{2} - 375251480 T^{3} + 259020342850 T^{4} - 102268282520348 T^{5} + 259020342850 p^{3} T^{6} - 375251480 p^{6} T^{7} + 689345 p^{9} T^{8} - 650 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 67 | \( 1 - 2016990 T^{2} + 1931325978405 T^{4} - 1175831601966849320 T^{6} + \)\(51\!\cdots\!30\)\( T^{8} - \)\(17\!\cdots\!32\)\( T^{10} + \)\(51\!\cdots\!30\)\( p^{6} T^{12} - 1175831601966849320 p^{12} T^{14} + 1931325978405 p^{18} T^{16} - 2016990 p^{24} T^{18} + p^{30} T^{20} \) | |
| 71 | \( ( 1 - 1470 T + 1881615 T^{2} - 1727909680 T^{3} + 1338943951990 T^{4} - 860023408325220 T^{5} + 1338943951990 p^{3} T^{6} - 1727909680 p^{6} T^{7} + 1881615 p^{9} T^{8} - 1470 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 73 | \( 1 - 2398886 T^{2} + 2442762516189 T^{4} - 1366613667006822216 T^{6} + \)\(48\!\cdots\!38\)\( T^{8} - \)\(16\!\cdots\!36\)\( T^{10} + \)\(48\!\cdots\!38\)\( p^{6} T^{12} - 1366613667006822216 p^{12} T^{14} + 2442762516189 p^{18} T^{16} - 2398886 p^{24} T^{18} + p^{30} T^{20} \) | |
| 79 | \( ( 1 - 820 T + 1736795 T^{2} - 1027136240 T^{3} + 1389341527610 T^{4} - 657402503393464 T^{5} + 1389341527610 p^{3} T^{6} - 1027136240 p^{6} T^{7} + 1736795 p^{9} T^{8} - 820 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 83 | \( 1 - 4743150 T^{2} + 10500593430085 T^{4} - 14340896860193237480 T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(90\!\cdots\!28\)\( T^{10} + \)\(13\!\cdots\!10\)\( p^{6} T^{12} - 14340896860193237480 p^{12} T^{14} + 10500593430085 p^{18} T^{16} - 4743150 p^{24} T^{18} + p^{30} T^{20} \) | |
| 89 | \( ( 1 + 286 T + 2177465 T^{2} + 600135040 T^{3} + 2162009088238 T^{4} + 560754412921604 T^{5} + 2162009088238 p^{3} T^{6} + 600135040 p^{6} T^{7} + 2177465 p^{9} T^{8} + 286 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 97 | \( 1 - 6782134 T^{2} + 21944276999789 T^{4} - 44851979434890226824 T^{6} + \)\(64\!\cdots\!18\)\( T^{8} - \)\(68\!\cdots\!04\)\( T^{10} + \)\(64\!\cdots\!18\)\( p^{6} T^{12} - 44851979434890226824 p^{12} T^{14} + 21944276999789 p^{18} T^{16} - 6782134 p^{24} T^{18} + p^{30} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.00718985857788251436213541895, −4.77370533576422905793775935756, −4.71700071546656835211690477987, −4.48577396528645633129518616264, −4.19181882194962807505477648783, −4.06723043474538779451258251349, −4.02165309549407933467306641639, −3.99974671043869865632450468572, −3.89359918634416439519060874551, −3.77129916583061970201418224630, −3.58427089350322792791823089277, −3.28246318049781549774943209161, −3.09883428944997569208472953217, −2.87180820998379020100398082014, −2.54522885983110857556360337603, −2.34324822840924792785866273889, −2.31979613775234766593493356426, −2.14837348286011675999983431706, −1.97318899501077797207547243183, −1.93752015487799232672000412063, −1.13786928987769514667714563043, −1.12900717035133115853167033658, −1.01861256683466620145631022893, −0.44928666203797719394675372730, −0.13608906542958302754975368007, 0.13608906542958302754975368007, 0.44928666203797719394675372730, 1.01861256683466620145631022893, 1.12900717035133115853167033658, 1.13786928987769514667714563043, 1.93752015487799232672000412063, 1.97318899501077797207547243183, 2.14837348286011675999983431706, 2.31979613775234766593493356426, 2.34324822840924792785866273889, 2.54522885983110857556360337603, 2.87180820998379020100398082014, 3.09883428944997569208472953217, 3.28246318049781549774943209161, 3.58427089350322792791823089277, 3.77129916583061970201418224630, 3.89359918634416439519060874551, 3.99974671043869865632450468572, 4.02165309549407933467306641639, 4.06723043474538779451258251349, 4.19181882194962807505477648783, 4.48577396528645633129518616264, 4.71700071546656835211690477987, 4.77370533576422905793775935756, 5.00718985857788251436213541895
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.