Dirichlet series
| L(s) = 1 | + 7·3-s + 4·7-s + 28·9-s + 4·11-s + 10·13-s + 6·17-s + 10·19-s + 28·21-s + 84·27-s + 10·29-s − 7·31-s + 28·33-s + 16·37-s + 70·39-s + 2·41-s − 8·43-s + 12·47-s − 12·49-s + 42·51-s + 16·53-s + 70·57-s + 14·59-s + 10·61-s + 112·63-s + 22·67-s − 2·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 4.04·3-s + 1.51·7-s + 28/3·9-s + 1.20·11-s + 2.77·13-s + 1.45·17-s + 2.29·19-s + 6.11·21-s + 16.1·27-s + 1.85·29-s − 1.25·31-s + 4.87·33-s + 2.63·37-s + 11.2·39-s + 0.312·41-s − 1.21·43-s + 1.75·47-s − 1.71·49-s + 5.88·51-s + 2.19·53-s + 9.27·57-s + 1.82·59-s + 1.28·61-s + 14.1·63-s + 2.68·67-s − 0.237·71-s + 1.63·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{7} \cdot 5^{14} \cdot 31^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{7} \cdot 5^{14} \cdot 31^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{14} \cdot 3^{7} \cdot 5^{14} \cdot 31^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.24543\times 10^{13}\) |
| Root analytic conductor: | \(8.61747\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{14} \cdot 3^{7} \cdot 5^{14} \cdot 31^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1546.853823\) |
| \(L(\frac12)\) | \(\approx\) | \(1546.853823\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{7} \) | |
| 5 | \( 1 \) | |
| 31 | \( ( 1 + T )^{7} \) | |
| good | 7 | \( 1 - 4 T + 4 p T^{2} - 82 T^{3} + 340 T^{4} - 760 T^{5} + 2721 T^{6} - 5298 T^{7} + 2721 p T^{8} - 760 p^{2} T^{9} + 340 p^{3} T^{10} - 82 p^{4} T^{11} + 4 p^{6} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) |
| 11 | \( 1 - 4 T + 49 T^{2} - 156 T^{3} + 1129 T^{4} - 2964 T^{5} + 16737 T^{6} - 37720 T^{7} + 16737 p T^{8} - 2964 p^{2} T^{9} + 1129 p^{3} T^{10} - 156 p^{4} T^{11} + 49 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 10 T + 89 T^{2} - 444 T^{3} + 2115 T^{4} - 6520 T^{5} + 1823 p T^{6} - 65188 T^{7} + 1823 p^{2} T^{8} - 6520 p^{2} T^{9} + 2115 p^{3} T^{10} - 444 p^{4} T^{11} + 89 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 6 T + 89 T^{2} - 428 T^{3} + 3787 T^{4} - 15046 T^{5} + 98059 T^{6} - 319680 T^{7} + 98059 p T^{8} - 15046 p^{2} T^{9} + 3787 p^{3} T^{10} - 428 p^{4} T^{11} + 89 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 10 T + 134 T^{2} - 928 T^{3} + 7204 T^{4} - 38214 T^{5} + 217363 T^{6} - 919152 T^{7} + 217363 p T^{8} - 38214 p^{2} T^{9} + 7204 p^{3} T^{10} - 928 p^{4} T^{11} + 134 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 107 T^{2} - 24 T^{3} + 5179 T^{4} - 1812 T^{5} + 159633 T^{6} - 57976 T^{7} + 159633 p T^{8} - 1812 p^{2} T^{9} + 5179 p^{3} T^{10} - 24 p^{4} T^{11} + 107 p^{5} T^{12} + p^{7} T^{14} \) | |
| 29 | \( 1 - 10 T + 147 T^{2} - 1036 T^{3} + 9399 T^{4} - 56464 T^{5} + 400365 T^{6} - 2038764 T^{7} + 400365 p T^{8} - 56464 p^{2} T^{9} + 9399 p^{3} T^{10} - 1036 p^{4} T^{11} + 147 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 16 T + 297 T^{2} - 3102 T^{3} + 33719 T^{4} - 261898 T^{5} + 2069439 T^{6} - 12506792 T^{7} + 2069439 p T^{8} - 261898 p^{2} T^{9} + 33719 p^{3} T^{10} - 3102 p^{4} T^{11} + 297 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 2 T + 140 T^{2} - 700 T^{3} + 10454 T^{4} - 60938 T^{5} + 622493 T^{6} - 2909464 T^{7} + 622493 p T^{8} - 60938 p^{2} T^{9} + 10454 p^{3} T^{10} - 700 p^{4} T^{11} + 140 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 8 T + 165 T^{2} + 1600 T^{3} + 16621 T^{4} + 133248 T^{5} + 1123969 T^{6} + 6920880 T^{7} + 1123969 p T^{8} + 133248 p^{2} T^{9} + 16621 p^{3} T^{10} + 1600 p^{4} T^{11} + 165 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 12 T + 287 T^{2} - 2208 T^{3} + 30043 T^{4} - 160968 T^{5} + 1776765 T^{6} - 7863080 T^{7} + 1776765 p T^{8} - 160968 p^{2} T^{9} + 30043 p^{3} T^{10} - 2208 p^{4} T^{11} + 287 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 16 T + 381 T^{2} - 4176 T^{3} + 57215 T^{4} - 479188 T^{5} + 4807699 T^{6} - 32111816 T^{7} + 4807699 p T^{8} - 479188 p^{2} T^{9} + 57215 p^{3} T^{10} - 4176 p^{4} T^{11} + 381 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 14 T + 350 T^{2} - 4272 T^{3} + 56374 T^{4} - 572388 T^{5} + 5325127 T^{6} - 43550822 T^{7} + 5325127 p T^{8} - 572388 p^{2} T^{9} + 56374 p^{3} T^{10} - 4272 p^{4} T^{11} + 350 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 10 T + 295 T^{2} - 2092 T^{3} + 39449 T^{4} - 218902 T^{5} + 3326087 T^{6} - 15332424 T^{7} + 3326087 p T^{8} - 218902 p^{2} T^{9} + 39449 p^{3} T^{10} - 2092 p^{4} T^{11} + 295 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 22 T + 383 T^{2} - 5000 T^{3} + 59075 T^{4} - 604800 T^{5} + 5665957 T^{6} - 47973132 T^{7} + 5665957 p T^{8} - 604800 p^{2} T^{9} + 59075 p^{3} T^{10} - 5000 p^{4} T^{11} + 383 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 2 T + 394 T^{2} + 718 T^{3} + 71850 T^{4} + 116274 T^{5} + 7868527 T^{6} + 10707690 T^{7} + 7868527 p T^{8} + 116274 p^{2} T^{9} + 71850 p^{3} T^{10} + 718 p^{4} T^{11} + 394 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 14 T + 437 T^{2} - 5394 T^{3} + 88755 T^{4} - 903184 T^{5} + 10502231 T^{6} - 85275512 T^{7} + 10502231 p T^{8} - 903184 p^{2} T^{9} + 88755 p^{3} T^{10} - 5394 p^{4} T^{11} + 437 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 24 T + 537 T^{2} - 8500 T^{3} + 118945 T^{4} - 1408612 T^{5} + 15125785 T^{6} - 140245776 T^{7} + 15125785 p T^{8} - 1408612 p^{2} T^{9} + 118945 p^{3} T^{10} - 8500 p^{4} T^{11} + 537 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 12 T + 391 T^{2} + 3972 T^{3} + 77835 T^{4} + 655880 T^{5} + 9641381 T^{6} + 68064960 T^{7} + 9641381 p T^{8} + 655880 p^{2} T^{9} + 77835 p^{3} T^{10} + 3972 p^{4} T^{11} + 391 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 12 T + 523 T^{2} - 4302 T^{3} + 112515 T^{4} - 679646 T^{5} + 14373785 T^{6} - 69937056 T^{7} + 14373785 p T^{8} - 679646 p^{2} T^{9} + 112515 p^{3} T^{10} - 4302 p^{4} T^{11} + 523 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 24 T + 464 T^{2} - 5472 T^{3} + 69910 T^{4} - 783016 T^{5} + 10268321 T^{6} - 103079560 T^{7} + 10268321 p T^{8} - 783016 p^{2} T^{9} + 69910 p^{3} T^{10} - 5472 p^{4} T^{11} + 464 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.42663511189813197836731717556, −3.35590004556614335289659272799, −3.35091999929945190338356172396, −3.33929058973293963660837564850, −2.89910485452864158553767327413, −2.78290646598152848629732580851, −2.65323139626905205866453424993, −2.64784280792535449639026643338, −2.61168978015938584919532612875, −2.60187208095619114802157971290, −2.38155019072095182969061301446, −1.87729787140411351739380614545, −1.87425999405026257645961406433, −1.85888754004708891732136215489, −1.79604597657324020753618060049, −1.72578379966646772260769470662, −1.67793735646034943631456642261, −1.58416238768447592616698865323, −1.08238789420473574257822552851, −0.957584095738522891211868265904, −0.904355668250851425416019684834, −0.795100760696013827644119652083, −0.74403157664167519036798737877, −0.70799979387566350948509083005, −0.63775399652097030718859771609, 0.63775399652097030718859771609, 0.70799979387566350948509083005, 0.74403157664167519036798737877, 0.795100760696013827644119652083, 0.904355668250851425416019684834, 0.957584095738522891211868265904, 1.08238789420473574257822552851, 1.58416238768447592616698865323, 1.67793735646034943631456642261, 1.72578379966646772260769470662, 1.79604597657324020753618060049, 1.85888754004708891732136215489, 1.87425999405026257645961406433, 1.87729787140411351739380614545, 2.38155019072095182969061301446, 2.60187208095619114802157971290, 2.61168978015938584919532612875, 2.64784280792535449639026643338, 2.65323139626905205866453424993, 2.78290646598152848629732580851, 2.89910485452864158553767327413, 3.33929058973293963660837564850, 3.35091999929945190338356172396, 3.35590004556614335289659272799, 3.42663511189813197836731717556
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.