Dirichlet series
| L(s) = 1 | + 4·2-s + 3-s + 5·4-s + 4·6-s + 7·7-s − 8-s − 6·9-s + 16·11-s + 5·12-s − 2·13-s + 28·14-s − 8·16-s + 4·17-s − 24·18-s + 7·21-s + 64·22-s + 6·23-s − 24-s − 19·25-s − 8·26-s − 11·27-s + 35·28-s + 12·29-s + 8·31-s − 7·32-s + 16·33-s + 16·34-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.577·3-s + 5/2·4-s + 1.63·6-s + 2.64·7-s − 0.353·8-s − 2·9-s + 4.82·11-s + 1.44·12-s − 0.554·13-s + 7.48·14-s − 2·16-s + 0.970·17-s − 5.65·18-s + 1.52·21-s + 13.6·22-s + 1.25·23-s − 0.204·24-s − 3.79·25-s − 1.56·26-s − 2.11·27-s + 6.61·28-s + 2.22·29-s + 1.43·31-s − 1.23·32-s + 2.78·33-s + 2.74·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{7} \cdot 43^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{7} \cdot 43^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(7^{7} \cdot 43^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(463.345\) |
| Root analytic conductor: | \(1.55032\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 7^{7} \cdot 43^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(22.98818426\) |
| \(L(\frac12)\) | \(\approx\) | \(22.98818426\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 - T )^{7} \) |
| 43 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 - p^{2} T + 11 T^{2} - 23 T^{3} + 41 T^{4} - 63 T^{5} + 93 T^{6} - 65 p T^{7} + 93 p T^{8} - 63 p^{2} T^{9} + 41 p^{3} T^{10} - 23 p^{4} T^{11} + 11 p^{5} T^{12} - p^{8} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - T + 7 T^{2} - 2 T^{3} + 22 T^{4} + p T^{5} + 16 p T^{6} + 32 T^{7} + 16 p^{2} T^{8} + p^{3} T^{9} + 22 p^{3} T^{10} - 2 p^{4} T^{11} + 7 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 + 19 T^{2} + 9 T^{3} + 182 T^{4} + 126 T^{5} + 1218 T^{6} + 826 T^{7} + 1218 p T^{8} + 126 p^{2} T^{9} + 182 p^{3} T^{10} + 9 p^{4} T^{11} + 19 p^{5} T^{12} + p^{7} T^{14} \) | |
| 11 | \( 1 - 16 T + 160 T^{2} - 1160 T^{3} + 6759 T^{4} - 32735 T^{5} + 135512 T^{6} - 482730 T^{7} + 135512 p T^{8} - 32735 p^{2} T^{9} + 6759 p^{3} T^{10} - 1160 p^{4} T^{11} + 160 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 2 T + 56 T^{2} + 54 T^{3} + 1283 T^{4} + p T^{5} + 18316 T^{6} - 698 p T^{7} + 18316 p T^{8} + p^{3} T^{9} + 1283 p^{3} T^{10} + 54 p^{4} T^{11} + 56 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 4 T + 90 T^{2} - 314 T^{3} + 3849 T^{4} - 11371 T^{5} + 99796 T^{6} - 244718 T^{7} + 99796 p T^{8} - 11371 p^{2} T^{9} + 3849 p^{3} T^{10} - 314 p^{4} T^{11} + 90 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 85 T^{2} - 87 T^{3} + 3206 T^{4} - 6430 T^{5} + 77298 T^{6} - 181558 T^{7} + 77298 p T^{8} - 6430 p^{2} T^{9} + 3206 p^{3} T^{10} - 87 p^{4} T^{11} + 85 p^{5} T^{12} + p^{7} T^{14} \) | |
| 23 | \( 1 - 6 T + 71 T^{2} - 250 T^{3} + 2127 T^{4} - 6423 T^{5} + 52401 T^{6} - 154626 T^{7} + 52401 p T^{8} - 6423 p^{2} T^{9} + 2127 p^{3} T^{10} - 250 p^{4} T^{11} + 71 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 12 T + 99 T^{2} - 718 T^{3} + 4840 T^{4} - 28786 T^{5} + 165816 T^{6} - 884280 T^{7} + 165816 p T^{8} - 28786 p^{2} T^{9} + 4840 p^{3} T^{10} - 718 p^{4} T^{11} + 99 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 8 T + 162 T^{2} - 1224 T^{3} + 12497 T^{4} - 84519 T^{5} + 590654 T^{6} - 3361598 T^{7} + 590654 p T^{8} - 84519 p^{2} T^{9} + 12497 p^{3} T^{10} - 1224 p^{4} T^{11} + 162 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 7 T + 97 T^{2} + 802 T^{3} + 7744 T^{4} + 49305 T^{5} + 394202 T^{6} + 2200172 T^{7} + 394202 p T^{8} + 49305 p^{2} T^{9} + 7744 p^{3} T^{10} + 802 p^{4} T^{11} + 97 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 14 T + 261 T^{2} - 2256 T^{3} + 25555 T^{4} - 170727 T^{5} + 1528351 T^{6} - 8481414 T^{7} + 1528351 p T^{8} - 170727 p^{2} T^{9} + 25555 p^{3} T^{10} - 2256 p^{4} T^{11} + 261 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - T + 231 T^{2} - 2 T^{3} + 24886 T^{4} + 12843 T^{5} + 1689648 T^{6} + 1010848 T^{7} + 1689648 p T^{8} + 12843 p^{2} T^{9} + 24886 p^{3} T^{10} - 2 p^{4} T^{11} + 231 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 15 T + 364 T^{2} - 4265 T^{3} + 56555 T^{4} - 528692 T^{5} + 4926696 T^{6} - 36542144 T^{7} + 4926696 p T^{8} - 528692 p^{2} T^{9} + 56555 p^{3} T^{10} - 4265 p^{4} T^{11} + 364 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 45 T + 1200 T^{2} - 22541 T^{3} + 330821 T^{4} - 3936453 T^{5} + 39059158 T^{6} - 326065954 T^{7} + 39059158 p T^{8} - 3936453 p^{2} T^{9} + 330821 p^{3} T^{10} - 22541 p^{4} T^{11} + 1200 p^{5} T^{12} - 45 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 20 T + 374 T^{2} + 4586 T^{3} + 52183 T^{4} + 498272 T^{5} + 4444126 T^{6} + 35976060 T^{7} + 4444126 p T^{8} + 498272 p^{2} T^{9} + 52183 p^{3} T^{10} + 4586 p^{4} T^{11} + 374 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 7 T + 364 T^{2} - 1955 T^{3} + 58901 T^{4} - 249424 T^{5} + 5782354 T^{6} - 20066356 T^{7} + 5782354 p T^{8} - 249424 p^{2} T^{9} + 58901 p^{3} T^{10} - 1955 p^{4} T^{11} + 364 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 13 T + 394 T^{2} - 4161 T^{3} + 72815 T^{4} - 631265 T^{5} + 8026890 T^{6} - 56769346 T^{7} + 8026890 p T^{8} - 631265 p^{2} T^{9} + 72815 p^{3} T^{10} - 4161 p^{4} T^{11} + 394 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 19 T + 466 T^{2} + 5093 T^{3} + 67959 T^{4} + 474225 T^{5} + 5016602 T^{6} + 29630694 T^{7} + 5016602 p T^{8} + 474225 p^{2} T^{9} + 67959 p^{3} T^{10} + 5093 p^{4} T^{11} + 466 p^{5} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 12 T + 252 T^{2} - 1578 T^{3} + 30157 T^{4} - 158760 T^{5} + 2854774 T^{6} - 11982532 T^{7} + 2854774 p T^{8} - 158760 p^{2} T^{9} + 30157 p^{3} T^{10} - 1578 p^{4} T^{11} + 252 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 6 T + 402 T^{2} + 1692 T^{3} + 71257 T^{4} + 206065 T^{5} + 7903742 T^{6} + 17715222 T^{7} + 7903742 p T^{8} + 206065 p^{2} T^{9} + 71257 p^{3} T^{10} + 1692 p^{4} T^{11} + 402 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 17 T + 528 T^{2} + 7371 T^{3} + 123275 T^{4} + 1435845 T^{5} + 16951108 T^{6} + 162682086 T^{7} + 16951108 p T^{8} + 1435845 p^{2} T^{9} + 123275 p^{3} T^{10} + 7371 p^{4} T^{11} + 528 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 15 T + 333 T^{2} + 3181 T^{3} + 30269 T^{4} + 148740 T^{5} - 248343 T^{6} - 4261096 T^{7} - 248343 p T^{8} + 148740 p^{2} T^{9} + 30269 p^{3} T^{10} + 3181 p^{4} T^{11} + 333 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−5.65161571917332461360873291550, −5.56149428288913732769856624732, −5.52510228592135476436621685445, −5.38535031536638533774483775231, −4.90946967738760821328125485673, −4.76275699173567542076742484592, −4.66997000543338756973345643248, −4.63957400875972780805929890740, −4.30753088778914360684944907983, −4.26835415600470484502782691082, −4.20437956772191626578110282692, −3.95141469068754697578589074226, −3.72017953175978764992954434935, −3.60639557020504476968686782104, −3.52441995182725366633079679588, −3.27502940086921209770329781370, −3.23749117520206814514257971575, −2.54826424830160291617475003502, −2.50749028558571409601400623175, −2.30446308741695073081884917481, −2.04138296252729884877785488820, −1.84216682171989952664369944239, −1.18170761347905843553320325391, −1.18071662845273145613049819135, −0.977793180312381191030459098669, 0.977793180312381191030459098669, 1.18071662845273145613049819135, 1.18170761347905843553320325391, 1.84216682171989952664369944239, 2.04138296252729884877785488820, 2.30446308741695073081884917481, 2.50749028558571409601400623175, 2.54826424830160291617475003502, 3.23749117520206814514257971575, 3.27502940086921209770329781370, 3.52441995182725366633079679588, 3.60639557020504476968686782104, 3.72017953175978764992954434935, 3.95141469068754697578589074226, 4.20437956772191626578110282692, 4.26835415600470484502782691082, 4.30753088778914360684944907983, 4.63957400875972780805929890740, 4.66997000543338756973345643248, 4.76275699173567542076742484592, 4.90946967738760821328125485673, 5.38535031536638533774483775231, 5.52510228592135476436621685445, 5.56149428288913732769856624732, 5.65161571917332461360873291550
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.