Dirichlet series
| L(s) = 1 | + 2·2-s + 5·3-s − 4-s + 11·5-s + 10·6-s + 4·7-s − 5·8-s + 4·9-s + 22·10-s + 8·11-s − 5·12-s − 7·13-s + 8·14-s + 55·15-s − 5·16-s + 7·17-s + 8·18-s + 19-s − 11·20-s + 20·21-s + 16·22-s + 6·23-s − 25·24-s + 48·25-s − 14·26-s − 23·27-s − 4·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.88·3-s − 1/2·4-s + 4.91·5-s + 4.08·6-s + 1.51·7-s − 1.76·8-s + 4/3·9-s + 6.95·10-s + 2.41·11-s − 1.44·12-s − 1.94·13-s + 2.13·14-s + 14.2·15-s − 5/4·16-s + 1.69·17-s + 1.88·18-s + 0.229·19-s − 2.45·20-s + 4.36·21-s + 3.41·22-s + 1.25·23-s − 5.10·24-s + 48/5·25-s − 2.74·26-s − 4.42·27-s − 0.755·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{7} \cdot 31^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{7} \cdot 31^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(13^{7} \cdot 31^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3573.34\) |
| Root analytic conductor: | \(1.79387\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 13^{7} \cdot 31^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(57.89560351\) |
| \(L(\frac12)\) | \(\approx\) | \(57.89560351\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 13 | \( ( 1 + T )^{7} \) |
| 31 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 - p T + 5 T^{2} - 7 T^{3} + 7 p T^{4} - 21 T^{5} + 41 T^{6} - 7 p^{3} T^{7} + 41 p T^{8} - 21 p^{2} T^{9} + 7 p^{4} T^{10} - 7 p^{4} T^{11} + 5 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - 5 T + 7 p T^{2} - 62 T^{3} + 164 T^{4} - 40 p^{2} T^{5} + 245 p T^{6} - 1306 T^{7} + 245 p^{2} T^{8} - 40 p^{4} T^{9} + 164 p^{3} T^{10} - 62 p^{4} T^{11} + 7 p^{6} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 - 11 T + 73 T^{2} - 357 T^{3} + 56 p^{2} T^{4} - 917 p T^{5} + 12789 T^{6} - 30754 T^{7} + 12789 p T^{8} - 917 p^{3} T^{9} + 56 p^{5} T^{10} - 357 p^{4} T^{11} + 73 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - 4 T + 6 p T^{2} - 144 T^{3} + 113 p T^{4} - 2291 T^{5} + 1245 p T^{6} - 20704 T^{7} + 1245 p^{2} T^{8} - 2291 p^{2} T^{9} + 113 p^{4} T^{10} - 144 p^{4} T^{11} + 6 p^{6} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 8 T + 74 T^{2} - 39 p T^{3} + 2386 T^{4} - 10471 T^{5} + 43268 T^{6} - 147557 T^{7} + 43268 p T^{8} - 10471 p^{2} T^{9} + 2386 p^{3} T^{10} - 39 p^{5} T^{11} + 74 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 7 T + 99 T^{2} - 487 T^{3} + 3936 T^{4} - 877 p T^{5} + 92493 T^{6} - 294386 T^{7} + 92493 p T^{8} - 877 p^{3} T^{9} + 3936 p^{3} T^{10} - 487 p^{4} T^{11} + 99 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - T + 96 T^{2} - 116 T^{3} + 4435 T^{4} - 5215 T^{5} + 127343 T^{6} - 128176 T^{7} + 127343 p T^{8} - 5215 p^{2} T^{9} + 4435 p^{3} T^{10} - 116 p^{4} T^{11} + 96 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 6 T + 85 T^{2} - 436 T^{3} + 3598 T^{4} - 15182 T^{5} + 100343 T^{6} - 386900 T^{7} + 100343 p T^{8} - 15182 p^{2} T^{9} + 3598 p^{3} T^{10} - 436 p^{4} T^{11} + 85 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 2 T + 95 T^{2} + 341 T^{3} + 4545 T^{4} + 22874 T^{5} + 158418 T^{6} + 847016 T^{7} + 158418 p T^{8} + 22874 p^{2} T^{9} + 4545 p^{3} T^{10} + 341 p^{4} T^{11} + 95 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 28 T + 543 T^{2} - 7259 T^{3} + 79366 T^{4} - 702049 T^{5} + 5368229 T^{6} - 34835635 T^{7} + 5368229 p T^{8} - 702049 p^{2} T^{9} + 79366 p^{3} T^{10} - 7259 p^{4} T^{11} + 543 p^{5} T^{12} - 28 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 3 T + 154 T^{2} - 384 T^{3} + 12246 T^{4} - 27106 T^{5} + 677120 T^{6} - 1349900 T^{7} + 677120 p T^{8} - 27106 p^{2} T^{9} + 12246 p^{3} T^{10} - 384 p^{4} T^{11} + 154 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + T + 182 T^{2} + 446 T^{3} + 15284 T^{4} + 56610 T^{5} + 839368 T^{6} + 3380702 T^{7} + 839368 p T^{8} + 56610 p^{2} T^{9} + 15284 p^{3} T^{10} + 446 p^{4} T^{11} + 182 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + T + 110 T^{2} + 628 T^{3} + 8078 T^{4} + 51096 T^{5} + 603016 T^{6} + 2275594 T^{7} + 603016 p T^{8} + 51096 p^{2} T^{9} + 8078 p^{3} T^{10} + 628 p^{4} T^{11} + 110 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 29 T + 591 T^{2} - 8451 T^{3} + 101858 T^{4} - 1016019 T^{5} + 9011455 T^{6} - 69125430 T^{7} + 9011455 p T^{8} - 1016019 p^{2} T^{9} + 101858 p^{3} T^{10} - 8451 p^{4} T^{11} + 591 p^{5} T^{12} - 29 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 3 T + 105 T^{2} - 407 T^{3} + 5420 T^{4} - 49707 T^{5} + 249413 T^{6} - 4514256 T^{7} + 249413 p T^{8} - 49707 p^{2} T^{9} + 5420 p^{3} T^{10} - 407 p^{4} T^{11} + 105 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 5 T + 315 T^{2} - 1450 T^{3} + 48188 T^{4} - 191734 T^{5} + 4491051 T^{6} - 14924570 T^{7} + 4491051 p T^{8} - 191734 p^{2} T^{9} + 48188 p^{3} T^{10} - 1450 p^{4} T^{11} + 315 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 32 T + 865 T^{2} + 15244 T^{3} + 234085 T^{4} + 2802431 T^{5} + 29745354 T^{6} + 257912236 T^{7} + 29745354 p T^{8} + 2802431 p^{2} T^{9} + 234085 p^{3} T^{10} + 15244 p^{4} T^{11} + 865 p^{5} T^{12} + 32 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 5 T + 232 T^{2} - 1756 T^{3} + 29821 T^{4} - 244839 T^{5} + 2745945 T^{6} - 21305042 T^{7} + 2745945 p T^{8} - 244839 p^{2} T^{9} + 29821 p^{3} T^{10} - 1756 p^{4} T^{11} + 232 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - T + 155 T^{2} + 1727 T^{3} + 10322 T^{4} + 214941 T^{5} + 2026687 T^{6} + 10857629 T^{7} + 2026687 p T^{8} + 214941 p^{2} T^{9} + 10322 p^{3} T^{10} + 1727 p^{4} T^{11} + 155 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 15 T + 496 T^{2} + 6030 T^{3} + 109559 T^{4} + 1086719 T^{5} + 13935127 T^{6} + 111068090 T^{7} + 13935127 p T^{8} + 1086719 p^{2} T^{9} + 109559 p^{3} T^{10} + 6030 p^{4} T^{11} + 496 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 17 T + 452 T^{2} - 5040 T^{3} + 77092 T^{4} - 623788 T^{5} + 7666358 T^{6} - 53524394 T^{7} + 7666358 p T^{8} - 623788 p^{2} T^{9} + 77092 p^{3} T^{10} - 5040 p^{4} T^{11} + 452 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 26 T + 730 T^{2} - 11864 T^{3} + 194429 T^{4} - 2319046 T^{5} + 27909218 T^{6} - 261497659 T^{7} + 27909218 p T^{8} - 2319046 p^{2} T^{9} + 194429 p^{3} T^{10} - 11864 p^{4} T^{11} + 730 p^{5} T^{12} - 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 11 T + 339 T^{2} - 1956 T^{3} + 40349 T^{4} - 18067 T^{5} + 2498592 T^{6} + 13581260 T^{7} + 2498592 p T^{8} - 18067 p^{2} T^{9} + 40349 p^{3} T^{10} - 1956 p^{4} T^{11} + 339 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−5.35732255581016432462348149862, −5.34591239953407421839771722474, −5.22065095825833563281602194168, −5.09717185831902291759656185891, −4.80486030495516029668815300900, −4.78662797687500769460929916336, −4.47234348554627009813807443360, −4.18721707464883420233967604042, −4.18690033350946735162270303109, −4.12668987108114259316458270442, −4.04514081658049784279356231606, −3.32394840403678948437393284703, −3.29408524822789261069591838065, −3.17587029915480142777935114071, −3.12560038356458798784862086808, −2.82601134736842073380180582416, −2.67418224012968921659978198168, −2.38455708021871366021232885516, −2.34223715586854073617878238859, −2.08649918029016325417391065340, −2.03133014690974556374267189730, −1.84577727613477550226818928670, −1.36156159798194282329633476310, −1.22041405406924334175042186781, −0.925480073843551150116958291249, 0.925480073843551150116958291249, 1.22041405406924334175042186781, 1.36156159798194282329633476310, 1.84577727613477550226818928670, 2.03133014690974556374267189730, 2.08649918029016325417391065340, 2.34223715586854073617878238859, 2.38455708021871366021232885516, 2.67418224012968921659978198168, 2.82601134736842073380180582416, 3.12560038356458798784862086808, 3.17587029915480142777935114071, 3.29408524822789261069591838065, 3.32394840403678948437393284703, 4.04514081658049784279356231606, 4.12668987108114259316458270442, 4.18690033350946735162270303109, 4.18721707464883420233967604042, 4.47234348554627009813807443360, 4.78662797687500769460929916336, 4.80486030495516029668815300900, 5.09717185831902291759656185891, 5.22065095825833563281602194168, 5.34591239953407421839771722474, 5.35732255581016432462348149862
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.