Dirichlet series
| L(s) = 1 | − 7·2-s − 7·3-s + 28·4-s + 2·5-s + 49·6-s − 9·7-s − 84·8-s + 28·9-s − 14·10-s − 196·12-s + 7·13-s + 63·14-s − 14·15-s + 210·16-s + 3·17-s − 196·18-s − 16·19-s + 56·20-s + 63·21-s + 6·23-s + 588·24-s − 8·25-s − 49·26-s − 84·27-s − 252·28-s − 5·29-s + 98·30-s + ⋯ |
| L(s) = 1 | − 4.94·2-s − 4.04·3-s + 14·4-s + 0.894·5-s + 20.0·6-s − 3.40·7-s − 29.6·8-s + 28/3·9-s − 4.42·10-s − 56.5·12-s + 1.94·13-s + 16.8·14-s − 3.61·15-s + 52.5·16-s + 0.727·17-s − 46.1·18-s − 3.67·19-s + 12.5·20-s + 13.7·21-s + 1.25·23-s + 120.·24-s − 8/5·25-s − 9.60·26-s − 16.1·27-s − 47.6·28-s − 0.928·29-s + 17.8·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 13^{7} \cdot 103^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 13^{7} \cdot 103^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{7} \cdot 3^{7} \cdot 13^{7} \cdot 103^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(4.47159\times 10^{12}\) |
| Root analytic conductor: | \(8.00948\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 2^{7} \cdot 3^{7} \cdot 13^{7} \cdot 103^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | 2 | \( ( 1 + T )^{7} \) |
| 3 | \( ( 1 + T )^{7} \) | |
| 13 | \( ( 1 - T )^{7} \) | |
| 103 | \( ( 1 - T )^{7} \) | |
| good | 5 | \( 1 - 2 T + 12 T^{2} - 19 T^{3} + 123 T^{4} - 209 T^{5} + 803 T^{6} - 983 T^{7} + 803 p T^{8} - 209 p^{2} T^{9} + 123 p^{3} T^{10} - 19 p^{4} T^{11} + 12 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 + 9 T + 66 T^{2} + 327 T^{3} + 1446 T^{4} + 5155 T^{5} + 16867 T^{6} + 46481 T^{7} + 16867 p T^{8} + 5155 p^{2} T^{9} + 1446 p^{3} T^{10} + 327 p^{4} T^{11} + 66 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 50 T^{2} - 5 T^{3} + 111 p T^{4} - 112 T^{5} + 19364 T^{6} - 1257 T^{7} + 19364 p T^{8} - 112 p^{2} T^{9} + 111 p^{4} T^{10} - 5 p^{4} T^{11} + 50 p^{5} T^{12} + p^{7} T^{14} \) | |
| 17 | \( 1 - 3 T + 48 T^{2} - 210 T^{3} + 1728 T^{4} - 6112 T^{5} + 41181 T^{6} - 130549 T^{7} + 41181 p T^{8} - 6112 p^{2} T^{9} + 1728 p^{3} T^{10} - 210 p^{4} T^{11} + 48 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 16 T + 225 T^{2} + 2028 T^{3} + 16324 T^{4} + 101598 T^{5} + 571970 T^{6} + 2616215 T^{7} + 571970 p T^{8} + 101598 p^{2} T^{9} + 16324 p^{3} T^{10} + 2028 p^{4} T^{11} + 225 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 6 T + 124 T^{2} - 579 T^{3} + 6515 T^{4} - 24568 T^{5} + 206736 T^{6} - 663561 T^{7} + 206736 p T^{8} - 24568 p^{2} T^{9} + 6515 p^{3} T^{10} - 579 p^{4} T^{11} + 124 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 5 T + 102 T^{2} + 612 T^{3} + 5141 T^{4} + 30066 T^{5} + 189821 T^{6} + 958991 T^{7} + 189821 p T^{8} + 30066 p^{2} T^{9} + 5141 p^{3} T^{10} + 612 p^{4} T^{11} + 102 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 16 T + 225 T^{2} + 2052 T^{3} + 592 p T^{4} + 128552 T^{5} + 886488 T^{6} + 4975541 T^{7} + 886488 p T^{8} + 128552 p^{2} T^{9} + 592 p^{4} T^{10} + 2052 p^{4} T^{11} + 225 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 17 T + 306 T^{2} - 3124 T^{3} + 32871 T^{4} - 245430 T^{5} + 1913427 T^{6} - 11334611 T^{7} + 1913427 p T^{8} - 245430 p^{2} T^{9} + 32871 p^{3} T^{10} - 3124 p^{4} T^{11} + 306 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 12 T + 4 p T^{2} - 964 T^{3} + 8573 T^{4} - 47076 T^{5} + 482781 T^{6} - 2693347 T^{7} + 482781 p T^{8} - 47076 p^{2} T^{9} + 8573 p^{3} T^{10} - 964 p^{4} T^{11} + 4 p^{6} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 22 T + 325 T^{2} + 3220 T^{3} + 24121 T^{4} + 136680 T^{5} + 650694 T^{6} + 3418661 T^{7} + 650694 p T^{8} + 136680 p^{2} T^{9} + 24121 p^{3} T^{10} + 3220 p^{4} T^{11} + 325 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 131 T^{2} - 16 T^{3} + 11688 T^{4} + 6954 T^{5} + 704218 T^{6} + 308813 T^{7} + 704218 p T^{8} + 6954 p^{2} T^{9} + 11688 p^{3} T^{10} - 16 p^{4} T^{11} + 131 p^{5} T^{12} + p^{7} T^{14} \) | |
| 53 | \( 1 - 2 T + 137 T^{2} + 393 T^{3} + 9678 T^{4} + 48589 T^{5} + 742820 T^{6} + 2496473 T^{7} + 742820 p T^{8} + 48589 p^{2} T^{9} + 9678 p^{3} T^{10} + 393 p^{4} T^{11} + 137 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 3 T + 210 T^{2} + 539 T^{3} + 25292 T^{4} + 53419 T^{5} + 2065249 T^{6} + 3896781 T^{7} + 2065249 p T^{8} + 53419 p^{2} T^{9} + 25292 p^{3} T^{10} + 539 p^{4} T^{11} + 210 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 6 T + 279 T^{2} + 1912 T^{3} + 39211 T^{4} + 258784 T^{5} + 3552046 T^{6} + 20060339 T^{7} + 3552046 p T^{8} + 258784 p^{2} T^{9} + 39211 p^{3} T^{10} + 1912 p^{4} T^{11} + 279 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - T + 211 T^{2} + 2 p T^{3} + 21876 T^{4} + 41069 T^{5} + 1677249 T^{6} + 3917111 T^{7} + 1677249 p T^{8} + 41069 p^{2} T^{9} + 21876 p^{3} T^{10} + 2 p^{5} T^{11} + 211 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 15 T + 452 T^{2} - 4851 T^{3} + 85045 T^{4} - 716014 T^{5} + 9325472 T^{6} - 63589473 T^{7} + 9325472 p T^{8} - 716014 p^{2} T^{9} + 85045 p^{3} T^{10} - 4851 p^{4} T^{11} + 452 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 17 T + 530 T^{2} - 6819 T^{3} + 117094 T^{4} - 1175660 T^{5} + 14247390 T^{6} - 112197599 T^{7} + 14247390 p T^{8} - 1175660 p^{2} T^{9} + 117094 p^{3} T^{10} - 6819 p^{4} T^{11} + 530 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 27 T + 744 T^{2} + 12364 T^{3} + 197835 T^{4} + 2372854 T^{5} + 27173877 T^{6} + 247348701 T^{7} + 27173877 p T^{8} + 2372854 p^{2} T^{9} + 197835 p^{3} T^{10} + 12364 p^{4} T^{11} + 744 p^{5} T^{12} + 27 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 12 T + 446 T^{2} - 4743 T^{3} + 96239 T^{4} - 859806 T^{5} + 12423630 T^{6} - 91258057 T^{7} + 12423630 p T^{8} - 859806 p^{2} T^{9} + 96239 p^{3} T^{10} - 4743 p^{4} T^{11} + 446 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 9 T + 340 T^{2} + 1689 T^{3} + 41782 T^{4} - 2472 T^{5} + 2633172 T^{6} - 14512419 T^{7} + 2633172 p T^{8} - 2472 p^{2} T^{9} + 41782 p^{3} T^{10} + 1689 p^{4} T^{11} + 340 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 3 T + 307 T^{2} + 1782 T^{3} + 44866 T^{4} + 386761 T^{5} + 4801989 T^{6} + 46932733 T^{7} + 4801989 p T^{8} + 386761 p^{2} T^{9} + 44866 p^{3} T^{10} + 1782 p^{4} T^{11} + 307 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.84697083873084914402180276043, −3.80768723178610052054885029856, −3.78255092591082478791654089450, −3.69467298131860357973060529329, −3.38756499587384812261273501490, −3.20797307415258400429063837591, −3.05069653403504079397863069218, −2.95321217729174614152826210791, −2.91116779127382566635657858161, −2.85440007280551662543352117777, −2.73361930513164286575816266526, −2.14840037383694094793760962160, −2.13847159063878761120244822571, −2.13309196936108115146503501238, −2.13215641186555478645179106778, −2.03852573706414966735884581159, −1.88550776225120894918549782352, −1.81844847898486280794005411894, −1.32894695600967012567916634250, −1.22779395800480222303085605842, −1.21204324084739057014127770493, −1.12598799712367628695757540994, −1.05626427956969767782349388641, −0.866625426477179672904931116611, −0.78019607539215726766331184144, 0, 0, 0, 0, 0, 0, 0, 0.78019607539215726766331184144, 0.866625426477179672904931116611, 1.05626427956969767782349388641, 1.12598799712367628695757540994, 1.21204324084739057014127770493, 1.22779395800480222303085605842, 1.32894695600967012567916634250, 1.81844847898486280794005411894, 1.88550776225120894918549782352, 2.03852573706414966735884581159, 2.13215641186555478645179106778, 2.13309196936108115146503501238, 2.13847159063878761120244822571, 2.14840037383694094793760962160, 2.73361930513164286575816266526, 2.85440007280551662543352117777, 2.91116779127382566635657858161, 2.95321217729174614152826210791, 3.05069653403504079397863069218, 3.20797307415258400429063837591, 3.38756499587384812261273501490, 3.69467298131860357973060529329, 3.78255092591082478791654089450, 3.80768723178610052054885029856, 3.84697083873084914402180276043
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.