Dirichlet series
| L(s) = 1 | − 13·2-s + 13·3-s + 91·4-s + 5-s − 169·6-s − 7-s − 455·8-s + 91·9-s − 13·10-s + 6·11-s + 1.18e3·12-s + 13·13-s + 13·14-s + 13·15-s + 1.82e3·16-s + 18·17-s − 1.18e3·18-s − 19-s + 91·20-s − 13·21-s − 78·22-s + 20·23-s − 5.91e3·24-s − 24·25-s − 169·26-s + 455·27-s − 91·28-s + ⋯ |
| L(s) = 1 | − 9.19·2-s + 7.50·3-s + 91/2·4-s + 0.447·5-s − 68.9·6-s − 0.377·7-s − 160.·8-s + 91/3·9-s − 4.11·10-s + 1.80·11-s + 341.·12-s + 3.60·13-s + 3.47·14-s + 3.35·15-s + 455·16-s + 4.36·17-s − 278.·18-s − 0.229·19-s + 20.3·20-s − 2.83·21-s − 16.6·22-s + 4.17·23-s − 1.20e3·24-s − 4.79·25-s − 33.1·26-s + 87.5·27-s − 17.1·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{13} \cdot 3^{13} \cdot 13^{13} \cdot 103^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{13} \cdot 3^{13} \cdot 13^{13} \cdot 103^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(2^{13} \cdot 3^{13} \cdot 13^{13} \cdot 103^{13}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.11685\times 10^{23}\) |
| Root analytic conductor: | \(8.00948\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((26,\ 2^{13} \cdot 3^{13} \cdot 13^{13} \cdot 103^{13} ,\ ( \ : [1/2]^{13} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(587.9546417\) |
| \(L(\frac12)\) | \(\approx\) | \(587.9546417\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{13} \) |
| 3 | \( ( 1 - T )^{13} \) | |
| 13 | \( ( 1 - T )^{13} \) | |
| 103 | \( ( 1 - T )^{13} \) | |
| good | 5 | \( 1 - T + p^{2} T^{2} - 26 T^{3} + 354 T^{4} - 331 T^{5} + 3578 T^{6} - 3016 T^{7} + 28964 T^{8} - 22223 T^{9} + 196931 T^{10} - 141731 T^{11} + 229303 p T^{12} - 768828 T^{13} + 229303 p^{2} T^{14} - 141731 p^{2} T^{15} + 196931 p^{3} T^{16} - 22223 p^{4} T^{17} + 28964 p^{5} T^{18} - 3016 p^{6} T^{19} + 3578 p^{7} T^{20} - 331 p^{8} T^{21} + 354 p^{9} T^{22} - 26 p^{10} T^{23} + p^{13} T^{24} - p^{12} T^{25} + p^{13} T^{26} \) |
| 7 | \( 1 + T + 46 T^{2} + 40 T^{3} + 1119 T^{4} + 823 T^{5} + 18792 T^{6} + 1689 p T^{7} + 241082 T^{8} + 132193 T^{9} + 2478482 T^{10} + 1206362 T^{11} + 2987911 p T^{12} + 9199108 T^{13} + 2987911 p^{2} T^{14} + 1206362 p^{2} T^{15} + 2478482 p^{3} T^{16} + 132193 p^{4} T^{17} + 241082 p^{5} T^{18} + 1689 p^{7} T^{19} + 18792 p^{7} T^{20} + 823 p^{8} T^{21} + 1119 p^{9} T^{22} + 40 p^{10} T^{23} + 46 p^{11} T^{24} + p^{12} T^{25} + p^{13} T^{26} \) | |
| 11 | \( 1 - 6 T + 95 T^{2} - 445 T^{3} + 4078 T^{4} - 15973 T^{5} + 111449 T^{6} - 383000 T^{7} + 2252090 T^{8} - 6966559 T^{9} + 36075172 T^{10} - 101525851 T^{11} + 474251051 T^{12} - 1221561420 T^{13} + 474251051 p T^{14} - 101525851 p^{2} T^{15} + 36075172 p^{3} T^{16} - 6966559 p^{4} T^{17} + 2252090 p^{5} T^{18} - 383000 p^{6} T^{19} + 111449 p^{7} T^{20} - 15973 p^{8} T^{21} + 4078 p^{9} T^{22} - 445 p^{10} T^{23} + 95 p^{11} T^{24} - 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 - 18 T + 271 T^{2} - 2833 T^{3} + 26652 T^{4} - 209033 T^{5} + 1518293 T^{6} - 9758968 T^{7} + 58951093 T^{8} - 323718808 T^{9} + 99076333 p T^{10} - 474832807 p T^{11} + 36809878173 T^{12} - 155489187802 T^{13} + 36809878173 p T^{14} - 474832807 p^{3} T^{15} + 99076333 p^{4} T^{16} - 323718808 p^{4} T^{17} + 58951093 p^{5} T^{18} - 9758968 p^{6} T^{19} + 1518293 p^{7} T^{20} - 209033 p^{8} T^{21} + 26652 p^{9} T^{22} - 2833 p^{10} T^{23} + 271 p^{11} T^{24} - 18 p^{12} T^{25} + p^{13} T^{26} \) | |
| 19 | \( 1 + T + 89 T^{2} + 118 T^{3} + 3551 T^{4} + 7307 T^{5} + 79541 T^{6} + 254589 T^{7} + 946180 T^{8} + 4707191 T^{9} + 979270 T^{10} + 33002165 T^{11} - 181609952 T^{12} - 63805734 T^{13} - 181609952 p T^{14} + 33002165 p^{2} T^{15} + 979270 p^{3} T^{16} + 4707191 p^{4} T^{17} + 946180 p^{5} T^{18} + 254589 p^{6} T^{19} + 79541 p^{7} T^{20} + 7307 p^{8} T^{21} + 3551 p^{9} T^{22} + 118 p^{10} T^{23} + 89 p^{11} T^{24} + p^{12} T^{25} + p^{13} T^{26} \) | |
| 23 | \( 1 - 20 T + 352 T^{2} - 4101 T^{3} + 1895 p T^{4} - 372518 T^{5} + 2979330 T^{6} - 20485645 T^{7} + 134510618 T^{8} - 786186384 T^{9} + 4479260759 T^{10} - 23276946973 T^{11} + 120274516911 T^{12} - 575825457774 T^{13} + 120274516911 p T^{14} - 23276946973 p^{2} T^{15} + 4479260759 p^{3} T^{16} - 786186384 p^{4} T^{17} + 134510618 p^{5} T^{18} - 20485645 p^{6} T^{19} + 2979330 p^{7} T^{20} - 372518 p^{8} T^{21} + 1895 p^{10} T^{22} - 4101 p^{10} T^{23} + 352 p^{11} T^{24} - 20 p^{12} T^{25} + p^{13} T^{26} \) | |
| 29 | \( 1 - 25 T + 514 T^{2} - 7329 T^{3} + 92318 T^{4} - 968258 T^{5} + 9295705 T^{6} - 79030818 T^{7} + 626852495 T^{8} - 4527137118 T^{9} + 30817131221 T^{10} - 193620860489 T^{11} + 1152727712381 T^{12} - 6371576750030 T^{13} + 1152727712381 p T^{14} - 193620860489 p^{2} T^{15} + 30817131221 p^{3} T^{16} - 4527137118 p^{4} T^{17} + 626852495 p^{5} T^{18} - 79030818 p^{6} T^{19} + 9295705 p^{7} T^{20} - 968258 p^{8} T^{21} + 92318 p^{9} T^{22} - 7329 p^{10} T^{23} + 514 p^{11} T^{24} - 25 p^{12} T^{25} + p^{13} T^{26} \) | |
| 31 | \( 1 + 5 T + 229 T^{2} + 1662 T^{3} + 27643 T^{4} + 228773 T^{5} + 2386285 T^{6} + 19044655 T^{7} + 157334264 T^{8} + 1121682255 T^{9} + 8005905466 T^{10} + 50136749907 T^{11} + 317045808232 T^{12} + 1751049030286 T^{13} + 317045808232 p T^{14} + 50136749907 p^{2} T^{15} + 8005905466 p^{3} T^{16} + 1121682255 p^{4} T^{17} + 157334264 p^{5} T^{18} + 19044655 p^{6} T^{19} + 2386285 p^{7} T^{20} + 228773 p^{8} T^{21} + 27643 p^{9} T^{22} + 1662 p^{10} T^{23} + 229 p^{11} T^{24} + 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 + 6 T + 229 T^{2} + 1563 T^{3} + 29319 T^{4} + 205714 T^{5} + 2622655 T^{6} + 18188219 T^{7} + 179633626 T^{8} + 1195429316 T^{9} + 9840351898 T^{10} + 61459463482 T^{11} + 440891702264 T^{12} + 2531076009400 T^{13} + 440891702264 p T^{14} + 61459463482 p^{2} T^{15} + 9840351898 p^{3} T^{16} + 1195429316 p^{4} T^{17} + 179633626 p^{5} T^{18} + 18188219 p^{6} T^{19} + 2622655 p^{7} T^{20} + 205714 p^{8} T^{21} + 29319 p^{9} T^{22} + 1563 p^{10} T^{23} + 229 p^{11} T^{24} + 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 - 13 T + 329 T^{2} - 4008 T^{3} + 57462 T^{4} - 611726 T^{5} + 6720724 T^{6} - 62188861 T^{7} + 572898395 T^{8} - 4677423501 T^{9} + 37417333446 T^{10} - 272037711237 T^{11} + 1925600799031 T^{12} - 12501862012612 T^{13} + 1925600799031 p T^{14} - 272037711237 p^{2} T^{15} + 37417333446 p^{3} T^{16} - 4677423501 p^{4} T^{17} + 572898395 p^{5} T^{18} - 62188861 p^{6} T^{19} + 6720724 p^{7} T^{20} - 611726 p^{8} T^{21} + 57462 p^{9} T^{22} - 4008 p^{10} T^{23} + 329 p^{11} T^{24} - 13 p^{12} T^{25} + p^{13} T^{26} \) | |
| 43 | \( 1 + 2 T + 227 T^{2} + 94 T^{3} + 29608 T^{4} - 10088 T^{5} + 2824117 T^{6} - 2157767 T^{7} + 211815474 T^{8} - 212412520 T^{9} + 12970645628 T^{10} - 14056618991 T^{11} + 663393251065 T^{12} - 691184010500 T^{13} + 663393251065 p T^{14} - 14056618991 p^{2} T^{15} + 12970645628 p^{3} T^{16} - 212412520 p^{4} T^{17} + 211815474 p^{5} T^{18} - 2157767 p^{6} T^{19} + 2824117 p^{7} T^{20} - 10088 p^{8} T^{21} + 29608 p^{9} T^{22} + 94 p^{10} T^{23} + 227 p^{11} T^{24} + 2 p^{12} T^{25} + p^{13} T^{26} \) | |
| 47 | \( 1 - 5 T + 286 T^{2} - 1338 T^{3} + 39347 T^{4} - 178291 T^{5} + 3533298 T^{6} - 15709377 T^{7} + 237763181 T^{8} - 1033499488 T^{9} + 13212651748 T^{10} - 55950858606 T^{11} + 658244739205 T^{12} - 2721293956630 T^{13} + 658244739205 p T^{14} - 55950858606 p^{2} T^{15} + 13212651748 p^{3} T^{16} - 1033499488 p^{4} T^{17} + 237763181 p^{5} T^{18} - 15709377 p^{6} T^{19} + 3533298 p^{7} T^{20} - 178291 p^{8} T^{21} + 39347 p^{9} T^{22} - 1338 p^{10} T^{23} + 286 p^{11} T^{24} - 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 53 | \( 1 - 21 T + 589 T^{2} - 8383 T^{3} + 136294 T^{4} - 1463302 T^{5} + 329963 p T^{6} - 148771630 T^{7} + 1441224347 T^{8} - 10024221099 T^{9} + 85365669693 T^{10} - 512392979211 T^{11} + 4298837598977 T^{12} - 25283227199172 T^{13} + 4298837598977 p T^{14} - 512392979211 p^{2} T^{15} + 85365669693 p^{3} T^{16} - 10024221099 p^{4} T^{17} + 1441224347 p^{5} T^{18} - 148771630 p^{6} T^{19} + 329963 p^{8} T^{20} - 1463302 p^{8} T^{21} + 136294 p^{9} T^{22} - 8383 p^{10} T^{23} + 589 p^{11} T^{24} - 21 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 - 22 T + 600 T^{2} - 9425 T^{3} + 151882 T^{4} - 1899182 T^{5} + 23244267 T^{6} - 248040679 T^{7} + 2554423688 T^{8} - 24346566537 T^{9} + 221972413919 T^{10} - 1927189113375 T^{11} + 15853218736929 T^{12} - 125386473045648 T^{13} + 15853218736929 p T^{14} - 1927189113375 p^{2} T^{15} + 221972413919 p^{3} T^{16} - 24346566537 p^{4} T^{17} + 2554423688 p^{5} T^{18} - 248040679 p^{6} T^{19} + 23244267 p^{7} T^{20} - 1899182 p^{8} T^{21} + 151882 p^{9} T^{22} - 9425 p^{10} T^{23} + 600 p^{11} T^{24} - 22 p^{12} T^{25} + p^{13} T^{26} \) | |
| 61 | \( 1 - 2 T + 295 T^{2} - 490 T^{3} + 48484 T^{4} - 81010 T^{5} + 5923599 T^{6} - 9592871 T^{7} + 583396486 T^{8} - 895865936 T^{9} + 48288553634 T^{10} - 70761375911 T^{11} + 3416338637165 T^{12} - 4697682281336 T^{13} + 3416338637165 p T^{14} - 70761375911 p^{2} T^{15} + 48288553634 p^{3} T^{16} - 895865936 p^{4} T^{17} + 583396486 p^{5} T^{18} - 9592871 p^{6} T^{19} + 5923599 p^{7} T^{20} - 81010 p^{8} T^{21} + 48484 p^{9} T^{22} - 490 p^{10} T^{23} + 295 p^{11} T^{24} - 2 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 + T + 325 T^{2} + 1023 T^{3} + 55755 T^{4} + 241106 T^{5} + 7121201 T^{6} + 32604747 T^{7} + 752311908 T^{8} + 3317852515 T^{9} + 67250045058 T^{10} + 286352503860 T^{11} + 5145893316345 T^{12} + 21040737104174 T^{13} + 5145893316345 p T^{14} + 286352503860 p^{2} T^{15} + 67250045058 p^{3} T^{16} + 3317852515 p^{4} T^{17} + 752311908 p^{5} T^{18} + 32604747 p^{6} T^{19} + 7121201 p^{7} T^{20} + 241106 p^{8} T^{21} + 55755 p^{9} T^{22} + 1023 p^{10} T^{23} + 325 p^{11} T^{24} + p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 - 32 T + 978 T^{2} - 20218 T^{3} + 383291 T^{4} - 6037364 T^{5} + 88023846 T^{6} - 1138407353 T^{7} + 13794584265 T^{8} - 152690668115 T^{9} + 1598671561773 T^{10} - 15523677790207 T^{11} + 143439830649220 T^{12} - 1237750583694574 T^{13} + 143439830649220 p T^{14} - 15523677790207 p^{2} T^{15} + 1598671561773 p^{3} T^{16} - 152690668115 p^{4} T^{17} + 13794584265 p^{5} T^{18} - 1138407353 p^{6} T^{19} + 88023846 p^{7} T^{20} - 6037364 p^{8} T^{21} + 383291 p^{9} T^{22} - 20218 p^{10} T^{23} + 978 p^{11} T^{24} - 32 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 + 13 T + 618 T^{2} + 7904 T^{3} + 195357 T^{4} + 2321074 T^{5} + 40912383 T^{6} + 440852563 T^{7} + 6245448502 T^{8} + 60624484621 T^{9} + 729440872702 T^{10} + 6366369654909 T^{11} + 66944930027678 T^{12} + 523353492943988 T^{13} + 66944930027678 p T^{14} + 6366369654909 p^{2} T^{15} + 729440872702 p^{3} T^{16} + 60624484621 p^{4} T^{17} + 6245448502 p^{5} T^{18} + 440852563 p^{6} T^{19} + 40912383 p^{7} T^{20} + 2321074 p^{8} T^{21} + 195357 p^{9} T^{22} + 7904 p^{10} T^{23} + 618 p^{11} T^{24} + 13 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 + 7 T + 666 T^{2} + 3966 T^{3} + 216212 T^{4} + 1126763 T^{5} + 45845413 T^{6} + 213787293 T^{7} + 7134900695 T^{8} + 30164548415 T^{9} + 863876465774 T^{10} + 3324058355493 T^{11} + 83908955488337 T^{12} + 292728058117086 T^{13} + 83908955488337 p T^{14} + 3324058355493 p^{2} T^{15} + 863876465774 p^{3} T^{16} + 30164548415 p^{4} T^{17} + 7134900695 p^{5} T^{18} + 213787293 p^{6} T^{19} + 45845413 p^{7} T^{20} + 1126763 p^{8} T^{21} + 216212 p^{9} T^{22} + 3966 p^{10} T^{23} + 666 p^{11} T^{24} + 7 p^{12} T^{25} + p^{13} T^{26} \) | |
| 83 | \( 1 - 17 T + 9 p T^{2} - 11982 T^{3} + 282330 T^{4} - 4121306 T^{5} + 70097218 T^{6} - 915823506 T^{7} + 12584642041 T^{8} - 146432891558 T^{9} + 1711928377951 T^{10} - 17730424063177 T^{11} + 180891835710606 T^{12} - 1665108893172028 T^{13} + 180891835710606 p T^{14} - 17730424063177 p^{2} T^{15} + 1711928377951 p^{3} T^{16} - 146432891558 p^{4} T^{17} + 12584642041 p^{5} T^{18} - 915823506 p^{6} T^{19} + 70097218 p^{7} T^{20} - 4121306 p^{8} T^{21} + 282330 p^{9} T^{22} - 11982 p^{10} T^{23} + 9 p^{12} T^{24} - 17 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 + 12 T + 531 T^{2} + 6016 T^{3} + 150579 T^{4} + 1541597 T^{5} + 29147971 T^{6} + 265730610 T^{7} + 4248135242 T^{8} + 34812908136 T^{9} + 500190450476 T^{10} + 3751238541798 T^{11} + 50320475683808 T^{12} + 352861386205326 T^{13} + 50320475683808 p T^{14} + 3751238541798 p^{2} T^{15} + 500190450476 p^{3} T^{16} + 34812908136 p^{4} T^{17} + 4248135242 p^{5} T^{18} + 265730610 p^{6} T^{19} + 29147971 p^{7} T^{20} + 1541597 p^{8} T^{21} + 150579 p^{9} T^{22} + 6016 p^{10} T^{23} + 531 p^{11} T^{24} + 12 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 + 42 T + 1908 T^{2} + 52434 T^{3} + 1409752 T^{4} + 29235853 T^{5} + 581942751 T^{6} + 9713128072 T^{7} + 154994147480 T^{8} + 2153837439916 T^{9} + 28605686577873 T^{10} + 337031300350094 T^{11} + 3796582535716027 T^{12} + 38237088196994506 T^{13} + 3796582535716027 p T^{14} + 337031300350094 p^{2} T^{15} + 28605686577873 p^{3} T^{16} + 2153837439916 p^{4} T^{17} + 154994147480 p^{5} T^{18} + 9713128072 p^{6} T^{19} + 581942751 p^{7} T^{20} + 29235853 p^{8} T^{21} + 1409752 p^{9} T^{22} + 52434 p^{10} T^{23} + 1908 p^{11} T^{24} + 42 p^{12} T^{25} + p^{13} T^{26} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.82197981365536666568429230304, −1.77744818463544824575736054740, −1.77575578765183966292813506430, −1.75301574390617812804569088490, −1.70997165198833344838152444365, −1.70457262574009312378176944755, −1.70402755759408013457804936502, −1.68527863001257833228000380875, −1.67826166350582869230390053057, −1.59268106663676917219662166721, −1.55988886050018290359245837889, −1.40422509190300029709754485344, −1.17992616183597063367825765493, −1.08238281958044933475655953299, −1.01662733190656866678937840984, −0.878055164205991727427518805918, −0.855037885004529546812495439673, −0.840954570638033187272913439027, −0.814459964848516427829760288692, −0.77907956562831183226219071087, −0.61483651948257900722242062610, −0.58738674129857564346682551079, −0.58461260600654690646747844994, −0.33553933816055386407542027941, −0.32757127489464671459079682747, 0.32757127489464671459079682747, 0.33553933816055386407542027941, 0.58461260600654690646747844994, 0.58738674129857564346682551079, 0.61483651948257900722242062610, 0.77907956562831183226219071087, 0.814459964848516427829760288692, 0.840954570638033187272913439027, 0.855037885004529546812495439673, 0.878055164205991727427518805918, 1.01662733190656866678937840984, 1.08238281958044933475655953299, 1.17992616183597063367825765493, 1.40422509190300029709754485344, 1.55988886050018290359245837889, 1.59268106663676917219662166721, 1.67826166350582869230390053057, 1.68527863001257833228000380875, 1.70402755759408013457804936502, 1.70457262574009312378176944755, 1.70997165198833344838152444365, 1.75301574390617812804569088490, 1.77575578765183966292813506430, 1.77744818463544824575736054740, 1.82197981365536666568429230304
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.