Dirichlet series
| L(s) = 1 | − 10·2-s − 10·3-s + 55·4-s + 6·5-s + 100·6-s − 9·7-s − 220·8-s + 55·9-s − 60·10-s − 11-s − 550·12-s − 10·13-s + 90·14-s − 60·15-s + 715·16-s + 5·17-s − 550·18-s − 9·19-s + 330·20-s + 90·21-s + 10·22-s + 23-s + 2.20e3·24-s + 3·25-s + 100·26-s − 220·27-s − 495·28-s + ⋯ |
| L(s) = 1 | − 7.07·2-s − 5.77·3-s + 55/2·4-s + 2.68·5-s + 40.8·6-s − 3.40·7-s − 77.7·8-s + 55/3·9-s − 18.9·10-s − 0.301·11-s − 158.·12-s − 2.77·13-s + 24.0·14-s − 15.4·15-s + 178.·16-s + 1.21·17-s − 129.·18-s − 2.06·19-s + 73.7·20-s + 19.6·21-s + 2.13·22-s + 0.208·23-s + 449.·24-s + 3/5·25-s + 19.6·26-s − 42.3·27-s − 93.5·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 13^{10} \cdot 103^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 13^{10} \cdot 103^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{10} \cdot 13^{10} \cdot 103^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.18056\times 10^{18}\) |
| Root analytic conductor: | \(8.00948\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(10\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{10} \cdot 13^{10} \cdot 103^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{10} \) |
| 3 | \( ( 1 + T )^{10} \) | |
| 13 | \( ( 1 + T )^{10} \) | |
| 103 | \( ( 1 + T )^{10} \) | |
| good | 5 | \( 1 - 6 T + 33 T^{2} - 119 T^{3} + 396 T^{4} - 212 p T^{5} + 2643 T^{6} - 5679 T^{7} + 11743 T^{8} - 22948 T^{9} + 49536 T^{10} - 22948 p T^{11} + 11743 p^{2} T^{12} - 5679 p^{3} T^{13} + 2643 p^{4} T^{14} - 212 p^{6} T^{15} + 396 p^{6} T^{16} - 119 p^{7} T^{17} + 33 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 + 9 T + 10 p T^{2} + 405 T^{3} + 2082 T^{4} + 9145 T^{5} + 36534 T^{6} + 130532 T^{7} + 429553 T^{8} + 1283988 T^{9} + 3556048 T^{10} + 1283988 p T^{11} + 429553 p^{2} T^{12} + 130532 p^{3} T^{13} + 36534 p^{4} T^{14} + 9145 p^{5} T^{15} + 2082 p^{6} T^{16} + 405 p^{7} T^{17} + 10 p^{9} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + T + 48 T^{2} + 30 T^{3} + 1164 T^{4} + 702 T^{5} + 19764 T^{6} + 15996 T^{7} + 268350 T^{8} + 266020 T^{9} + 3140003 T^{10} + 266020 p T^{11} + 268350 p^{2} T^{12} + 15996 p^{3} T^{13} + 19764 p^{4} T^{14} + 702 p^{5} T^{15} + 1164 p^{6} T^{16} + 30 p^{7} T^{17} + 48 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 5 T + 127 T^{2} - 617 T^{3} + 7601 T^{4} - 35257 T^{5} + 284942 T^{6} - 1236768 T^{7} + 7502944 T^{8} - 29589705 T^{9} + 146715906 T^{10} - 29589705 p T^{11} + 7502944 p^{2} T^{12} - 1236768 p^{3} T^{13} + 284942 p^{4} T^{14} - 35257 p^{5} T^{15} + 7601 p^{6} T^{16} - 617 p^{7} T^{17} + 127 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 9 T + 132 T^{2} + 825 T^{3} + 7594 T^{4} + 38611 T^{5} + 283212 T^{6} + 1249500 T^{7} + 7869398 T^{8} + 30764353 T^{9} + 169590490 T^{10} + 30764353 p T^{11} + 7869398 p^{2} T^{12} + 1249500 p^{3} T^{13} + 283212 p^{4} T^{14} + 38611 p^{5} T^{15} + 7594 p^{6} T^{16} + 825 p^{7} T^{17} + 132 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - T + 76 T^{2} - 194 T^{3} + 3802 T^{4} - 11252 T^{5} + 6400 p T^{6} - 440492 T^{7} + 4566628 T^{8} - 12582908 T^{9} + 117418651 T^{10} - 12582908 p T^{11} + 4566628 p^{2} T^{12} - 440492 p^{3} T^{13} + 6400 p^{5} T^{14} - 11252 p^{5} T^{15} + 3802 p^{6} T^{16} - 194 p^{7} T^{17} + 76 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 22 T + 386 T^{2} + 4813 T^{3} + 52836 T^{4} + 487019 T^{5} + 4074064 T^{6} + 30067703 T^{7} + 204295736 T^{8} + 1249154483 T^{9} + 7072911694 T^{10} + 1249154483 p T^{11} + 204295736 p^{2} T^{12} + 30067703 p^{3} T^{13} + 4074064 p^{4} T^{14} + 487019 p^{5} T^{15} + 52836 p^{6} T^{16} + 4813 p^{7} T^{17} + 386 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 13 T + 276 T^{2} + 2731 T^{3} + 33920 T^{4} + 272263 T^{5} + 2522336 T^{6} + 16989876 T^{7} + 127708332 T^{8} + 733254463 T^{9} + 4638493462 T^{10} + 733254463 p T^{11} + 127708332 p^{2} T^{12} + 16989876 p^{3} T^{13} + 2522336 p^{4} T^{14} + 272263 p^{5} T^{15} + 33920 p^{6} T^{16} + 2731 p^{7} T^{17} + 276 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 10 T + 161 T^{2} - 1449 T^{3} + 17204 T^{4} - 130918 T^{5} + 1227005 T^{6} - 8351458 T^{7} + 66380764 T^{8} - 399331867 T^{9} + 2782931033 T^{10} - 399331867 p T^{11} + 66380764 p^{2} T^{12} - 8351458 p^{3} T^{13} + 1227005 p^{4} T^{14} - 130918 p^{5} T^{15} + 17204 p^{6} T^{16} - 1449 p^{7} T^{17} + 161 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 18 T + 310 T^{2} + 4050 T^{3} + 46843 T^{4} + 470334 T^{5} + 4369368 T^{6} + 36294369 T^{7} + 282692915 T^{8} + 2012129172 T^{9} + 13424509742 T^{10} + 2012129172 p T^{11} + 282692915 p^{2} T^{12} + 36294369 p^{3} T^{13} + 4369368 p^{4} T^{14} + 470334 p^{5} T^{15} + 46843 p^{6} T^{16} + 4050 p^{7} T^{17} + 310 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 10 T + 145 T^{2} - 644 T^{3} + 6903 T^{4} - 12742 T^{5} + 314959 T^{6} - 735697 T^{7} + 20964427 T^{8} - 69161616 T^{9} + 1102849852 T^{10} - 69161616 p T^{11} + 20964427 p^{2} T^{12} - 735697 p^{3} T^{13} + 314959 p^{4} T^{14} - 12742 p^{5} T^{15} + 6903 p^{6} T^{16} - 644 p^{7} T^{17} + 145 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 28 T + 525 T^{2} - 7082 T^{3} + 79782 T^{4} - 755318 T^{5} + 6407173 T^{6} - 48902745 T^{7} + 353906913 T^{8} - 2447707186 T^{9} + 16907175154 T^{10} - 2447707186 p T^{11} + 353906913 p^{2} T^{12} - 48902745 p^{3} T^{13} + 6407173 p^{4} T^{14} - 755318 p^{5} T^{15} + 79782 p^{6} T^{16} - 7082 p^{7} T^{17} + 525 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 6 T + 148 T^{2} - 577 T^{3} + 18122 T^{4} - 88084 T^{5} + 1612384 T^{6} - 6355685 T^{7} + 109954477 T^{8} - 454117300 T^{9} + 6685717000 T^{10} - 454117300 p T^{11} + 109954477 p^{2} T^{12} - 6355685 p^{3} T^{13} + 1612384 p^{4} T^{14} - 88084 p^{5} T^{15} + 18122 p^{6} T^{16} - 577 p^{7} T^{17} + 148 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 7 T + 355 T^{2} - 1808 T^{3} + 57695 T^{4} - 208921 T^{5} + 6009928 T^{6} - 15359228 T^{7} + 473347084 T^{8} - 921913776 T^{9} + 30539394946 T^{10} - 921913776 p T^{11} + 473347084 p^{2} T^{12} - 15359228 p^{3} T^{13} + 6009928 p^{4} T^{14} - 208921 p^{5} T^{15} + 57695 p^{6} T^{16} - 1808 p^{7} T^{17} + 355 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 20 T + 441 T^{2} + 5710 T^{3} + 78769 T^{4} + 811298 T^{5} + 8939709 T^{6} + 80247589 T^{7} + 768632771 T^{8} + 6179624980 T^{9} + 52506357802 T^{10} + 6179624980 p T^{11} + 768632771 p^{2} T^{12} + 80247589 p^{3} T^{13} + 8939709 p^{4} T^{14} + 811298 p^{5} T^{15} + 78769 p^{6} T^{16} + 5710 p^{7} T^{17} + 441 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 21 T + 568 T^{2} + 7721 T^{3} + 120514 T^{4} + 1192978 T^{5} + 13803811 T^{6} + 106576787 T^{7} + 1058044567 T^{8} + 7083060761 T^{9} + 70142689590 T^{10} + 7083060761 p T^{11} + 1058044567 p^{2} T^{12} + 106576787 p^{3} T^{13} + 13803811 p^{4} T^{14} + 1192978 p^{5} T^{15} + 120514 p^{6} T^{16} + 7721 p^{7} T^{17} + 568 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 19 T + 615 T^{2} + 9516 T^{3} + 175586 T^{4} + 2249534 T^{5} + 30631826 T^{6} + 330136525 T^{7} + 3619007325 T^{8} + 33115661852 T^{9} + 303109809166 T^{10} + 33115661852 p T^{11} + 3619007325 p^{2} T^{12} + 330136525 p^{3} T^{13} + 30631826 p^{4} T^{14} + 2249534 p^{5} T^{15} + 175586 p^{6} T^{16} + 9516 p^{7} T^{17} + 615 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 3 T + 309 T^{2} - 716 T^{3} + 47073 T^{4} - 130422 T^{5} + 5114547 T^{6} - 20333915 T^{7} + 451213022 T^{8} - 2182565660 T^{9} + 34594985936 T^{10} - 2182565660 p T^{11} + 451213022 p^{2} T^{12} - 20333915 p^{3} T^{13} + 5114547 p^{4} T^{14} - 130422 p^{5} T^{15} + 47073 p^{6} T^{16} - 716 p^{7} T^{17} + 309 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 11 T + 429 T^{2} + 5669 T^{3} + 96628 T^{4} + 1288915 T^{5} + 15576599 T^{6} + 179207040 T^{7} + 1910082173 T^{8} + 17951900075 T^{9} + 175373846420 T^{10} + 17951900075 p T^{11} + 1910082173 p^{2} T^{12} + 179207040 p^{3} T^{13} + 15576599 p^{4} T^{14} + 1288915 p^{5} T^{15} + 96628 p^{6} T^{16} + 5669 p^{7} T^{17} + 429 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 20 T + 665 T^{2} - 10583 T^{3} + 206252 T^{4} - 2717547 T^{5} + 39755122 T^{6} - 443500612 T^{7} + 5304149359 T^{8} - 50680008772 T^{9} + 513349859410 T^{10} - 50680008772 p T^{11} + 5304149359 p^{2} T^{12} - 443500612 p^{3} T^{13} + 39755122 p^{4} T^{14} - 2717547 p^{5} T^{15} + 206252 p^{6} T^{16} - 10583 p^{7} T^{17} + 665 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 22 T + 794 T^{2} - 12610 T^{3} + 265034 T^{4} - 3350801 T^{5} + 52699890 T^{6} - 559879199 T^{7} + 7246733328 T^{8} - 66580491672 T^{9} + 740261461294 T^{10} - 66580491672 p T^{11} + 7246733328 p^{2} T^{12} - 559879199 p^{3} T^{13} + 52699890 p^{4} T^{14} - 3350801 p^{5} T^{15} + 265034 p^{6} T^{16} - 12610 p^{7} T^{17} + 794 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 10 T + 751 T^{2} + 6378 T^{3} + 267320 T^{4} + 1972103 T^{5} + 59788995 T^{6} + 385449599 T^{7} + 9305486822 T^{8} + 52173647406 T^{9} + 1052313453378 T^{10} + 52173647406 p T^{11} + 9305486822 p^{2} T^{12} + 385449599 p^{3} T^{13} + 59788995 p^{4} T^{14} + 1972103 p^{5} T^{15} + 267320 p^{6} T^{16} + 6378 p^{7} T^{17} + 751 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−2.87601607983744077863447590557, −2.86440448330526841417721886344, −2.78847784130197712027339842667, −2.62785816698126487349260220144, −2.60961374137913152792609410503, −2.44542128339835244499891040466, −2.18412568188417771410019157623, −2.16981748092747386188269128528, −2.08376673136693648188962794780, −2.08069407542178086001755486707, −2.03888111209899837568691444773, −2.03774253418648711855374471551, −1.92468346032399858774069431573, −1.86134595869135273821497558282, −1.77138244117482798440757826176, −1.50675007339327076443603491070, −1.32668120595143556679473235976, −1.26842616154523329745994630319, −1.23103504121739320786485153486, −1.20925607585711618526468635167, −1.11784762162036246495767798716, −1.01658866575865873078021936617, −0.986519549946495541230958889013, −0.793471101565537211818708497129, −0.64238703580943644630865332956, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.64238703580943644630865332956, 0.793471101565537211818708497129, 0.986519549946495541230958889013, 1.01658866575865873078021936617, 1.11784762162036246495767798716, 1.20925607585711618526468635167, 1.23103504121739320786485153486, 1.26842616154523329745994630319, 1.32668120595143556679473235976, 1.50675007339327076443603491070, 1.77138244117482798440757826176, 1.86134595869135273821497558282, 1.92468346032399858774069431573, 2.03774253418648711855374471551, 2.03888111209899837568691444773, 2.08069407542178086001755486707, 2.08376673136693648188962794780, 2.16981748092747386188269128528, 2.18412568188417771410019157623, 2.44542128339835244499891040466, 2.60961374137913152792609410503, 2.62785816698126487349260220144, 2.78847784130197712027339842667, 2.86440448330526841417721886344, 2.87601607983744077863447590557
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.