Dirichlet series
| L(s) = 1 | − 5·2-s + 3-s + 9·4-s − 4·5-s − 5·6-s − 8·7-s − 4·8-s − 9·9-s + 20·10-s − 13·11-s + 9·12-s − 7·13-s + 40·14-s − 4·15-s − 8·16-s + 2·17-s + 45·18-s − 8·19-s − 36·20-s − 8·21-s + 65·22-s − 18·23-s − 4·24-s − 6·25-s + 35·26-s − 8·27-s − 72·28-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 0.577·3-s + 9/2·4-s − 1.78·5-s − 2.04·6-s − 3.02·7-s − 1.41·8-s − 3·9-s + 6.32·10-s − 3.91·11-s + 2.59·12-s − 1.94·13-s + 10.6·14-s − 1.03·15-s − 2·16-s + 0.485·17-s + 10.6·18-s − 1.83·19-s − 8.04·20-s − 1.74·21-s + 13.8·22-s − 3.75·23-s − 0.816·24-s − 6/5·25-s + 6.86·26-s − 1.53·27-s − 13.6·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{7} \cdot 37^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{7} \cdot 37^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(13^{7} \cdot 37^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(12329.7\) |
| Root analytic conductor: | \(1.95979\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 13^{7} \cdot 37^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | 13 | \( ( 1 + T )^{7} \) |
| 37 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 + 5 T + p^{4} T^{2} + 39 T^{3} + 79 T^{4} + 35 p^{2} T^{5} + 223 T^{6} + 163 p T^{7} + 223 p T^{8} + 35 p^{4} T^{9} + 79 p^{3} T^{10} + 39 p^{4} T^{11} + p^{9} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - T + 10 T^{2} - 11 T^{3} + 7 p^{2} T^{4} - 19 p T^{5} + 260 T^{6} - 217 T^{7} + 260 p T^{8} - 19 p^{3} T^{9} + 7 p^{5} T^{10} - 11 p^{4} T^{11} + 10 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 + 4 T + 22 T^{2} + 78 T^{3} + 283 T^{4} + 759 T^{5} + 2143 T^{6} + 4784 T^{7} + 2143 p T^{8} + 759 p^{2} T^{9} + 283 p^{3} T^{10} + 78 p^{4} T^{11} + 22 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 8 T + 44 T^{2} + 27 p T^{3} + 104 p T^{4} + 2385 T^{5} + 7314 T^{6} + 20113 T^{7} + 7314 p T^{8} + 2385 p^{2} T^{9} + 104 p^{4} T^{10} + 27 p^{5} T^{11} + 44 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 13 T + 115 T^{2} + 775 T^{3} + 4204 T^{4} + 19509 T^{5} + 78403 T^{6} + 276291 T^{7} + 78403 p T^{8} + 19509 p^{2} T^{9} + 4204 p^{3} T^{10} + 775 p^{4} T^{11} + 115 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 2 T + 98 T^{2} - 176 T^{3} + 4413 T^{4} - 6837 T^{5} + 117619 T^{6} - 150454 T^{7} + 117619 p T^{8} - 6837 p^{2} T^{9} + 4413 p^{3} T^{10} - 176 p^{4} T^{11} + 98 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 8 T + 82 T^{2} + 351 T^{3} + 2528 T^{4} + 9931 T^{5} + 70962 T^{6} + 13408 p T^{7} + 70962 p T^{8} + 9931 p^{2} T^{9} + 2528 p^{3} T^{10} + 351 p^{4} T^{11} + 82 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 18 T + 233 T^{2} + 2096 T^{3} + 16225 T^{4} + 104773 T^{5} + 611406 T^{6} + 3082840 T^{7} + 611406 p T^{8} + 104773 p^{2} T^{9} + 16225 p^{3} T^{10} + 2096 p^{4} T^{11} + 233 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 9 T + 171 T^{2} + 1322 T^{3} + 13163 T^{4} + 86119 T^{5} + 595752 T^{6} + 3210634 T^{7} + 595752 p T^{8} + 86119 p^{2} T^{9} + 13163 p^{3} T^{10} + 1322 p^{4} T^{11} + 171 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 25 T + 420 T^{2} + 5056 T^{3} + 49651 T^{4} + 403515 T^{5} + 2812147 T^{6} + 16809144 T^{7} + 2812147 p T^{8} + 403515 p^{2} T^{9} + 49651 p^{3} T^{10} + 5056 p^{4} T^{11} + 420 p^{5} T^{12} + 25 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 7 T + 183 T^{2} - 1202 T^{3} + 17853 T^{4} - 99546 T^{5} + 1086113 T^{6} - 5128131 T^{7} + 1086113 p T^{8} - 99546 p^{2} T^{9} + 17853 p^{3} T^{10} - 1202 p^{4} T^{11} + 183 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 3 T + 175 T^{2} + 664 T^{3} + 15839 T^{4} + 64583 T^{5} + 952552 T^{6} + 3568882 T^{7} + 952552 p T^{8} + 64583 p^{2} T^{9} + 15839 p^{3} T^{10} + 664 p^{4} T^{11} + 175 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 3 T + 174 T^{2} + 855 T^{3} + 15878 T^{4} + 96439 T^{5} + 979730 T^{6} + 5956345 T^{7} + 979730 p T^{8} + 96439 p^{2} T^{9} + 15878 p^{3} T^{10} + 855 p^{4} T^{11} + 174 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 4 T + 92 T^{2} + 523 T^{3} + 178 p T^{4} + 38511 T^{5} + 587314 T^{6} + 2355111 T^{7} + 587314 p T^{8} + 38511 p^{2} T^{9} + 178 p^{4} T^{10} + 523 p^{4} T^{11} + 92 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 26 T + 454 T^{2} + 6054 T^{3} + 66961 T^{4} + 657885 T^{5} + 5868013 T^{6} + 46967158 T^{7} + 5868013 p T^{8} + 657885 p^{2} T^{9} + 66961 p^{3} T^{10} + 6054 p^{4} T^{11} + 454 p^{5} T^{12} + 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 8 T + 313 T^{2} - 2066 T^{3} + 46917 T^{4} - 260145 T^{5} + 4336022 T^{6} - 19942850 T^{7} + 4336022 p T^{8} - 260145 p^{2} T^{9} + 46917 p^{3} T^{10} - 2066 p^{4} T^{11} + 313 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 5 T + 342 T^{2} - 1210 T^{3} + 814 p T^{4} - 147108 T^{5} + 5416318 T^{6} - 11815970 T^{7} + 5416318 p T^{8} - 147108 p^{2} T^{9} + 814 p^{4} T^{10} - 1210 p^{4} T^{11} + 342 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 61 T + 1895 T^{2} + 40011 T^{3} + 643520 T^{4} + 8346295 T^{5} + 90081673 T^{6} + 822837541 T^{7} + 90081673 p T^{8} + 8346295 p^{2} T^{9} + 643520 p^{3} T^{10} + 40011 p^{4} T^{11} + 1895 p^{5} T^{12} + 61 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 14 T + 366 T^{2} + 3369 T^{3} + 53094 T^{4} + 363643 T^{5} + 4825684 T^{6} + 28319571 T^{7} + 4825684 p T^{8} + 363643 p^{2} T^{9} + 53094 p^{3} T^{10} + 3369 p^{4} T^{11} + 366 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 12 T + 354 T^{2} + 2659 T^{3} + 47026 T^{4} + 207919 T^{5} + 3612372 T^{6} + 11388664 T^{7} + 3612372 p T^{8} + 207919 p^{2} T^{9} + 47026 p^{3} T^{10} + 2659 p^{4} T^{11} + 354 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 5 T + 153 T^{2} - 1136 T^{3} + 22086 T^{4} - 100746 T^{5} + 1915351 T^{6} - 11954199 T^{7} + 1915351 p T^{8} - 100746 p^{2} T^{9} + 22086 p^{3} T^{10} - 1136 p^{4} T^{11} + 153 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 31 T + 706 T^{2} + 11300 T^{3} + 159605 T^{4} + 1943097 T^{5} + 21933461 T^{6} + 216806116 T^{7} + 21933461 p T^{8} + 1943097 p^{2} T^{9} + 159605 p^{3} T^{10} + 11300 p^{4} T^{11} + 706 p^{5} T^{12} + 31 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 2 T - 29 T^{2} + 874 T^{3} + 7403 T^{4} - 25435 T^{5} + 579454 T^{6} + 1966070 T^{7} + 579454 p T^{8} - 25435 p^{2} T^{9} + 7403 p^{3} T^{10} + 874 p^{4} T^{11} - 29 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−5.79449220106059188920365511807, −5.77839010391255944990456887926, −5.75897626677793218803738386798, −5.71063232516092451287889743889, −5.47528192641377294337600847284, −5.02091209530791765691013051361, −4.94549287446285080434224648213, −4.76464689893171505500115893350, −4.67783252515727957088299993860, −4.35494945194262757704122332472, −4.24206130742351484220671149988, −3.92221366662423302521757298314, −3.78819676172630869670599249556, −3.71045255908317966949024433670, −3.47634846261217546798943462065, −3.24069182090155425166225651640, −3.18727561873737037763747489449, −3.03348091756902915794480086628, −2.77919090560497571127401542267, −2.71416020861448242365418751342, −2.33026583916995233394720772716, −2.21822098237094906784488449268, −1.98725367777371258377506040318, −1.83497091798559391326210434061, −1.52807035536028366453061040954, 0, 0, 0, 0, 0, 0, 0, 1.52807035536028366453061040954, 1.83497091798559391326210434061, 1.98725367777371258377506040318, 2.21822098237094906784488449268, 2.33026583916995233394720772716, 2.71416020861448242365418751342, 2.77919090560497571127401542267, 3.03348091756902915794480086628, 3.18727561873737037763747489449, 3.24069182090155425166225651640, 3.47634846261217546798943462065, 3.71045255908317966949024433670, 3.78819676172630869670599249556, 3.92221366662423302521757298314, 4.24206130742351484220671149988, 4.35494945194262757704122332472, 4.67783252515727957088299993860, 4.76464689893171505500115893350, 4.94549287446285080434224648213, 5.02091209530791765691013051361, 5.47528192641377294337600847284, 5.71063232516092451287889743889, 5.75897626677793218803738386798, 5.77839010391255944990456887926, 5.79449220106059188920365511807
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.