Dirichlet series
| L(s) = 1 | − 4·2-s + 8·3-s + 6·4-s + 2·5-s − 32·6-s − 3·7-s + 36·9-s − 8·10-s − 8·11-s + 48·12-s + 3·13-s + 12·14-s + 16·15-s − 15·16-s − 2·17-s − 144·18-s + 8·19-s + 12·20-s − 24·21-s + 32·22-s + 4·23-s + 13·25-s − 12·26-s + 120·27-s − 18·28-s + 2·29-s − 64·30-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4.61·3-s + 3·4-s + 0.894·5-s − 13.0·6-s − 1.13·7-s + 12·9-s − 2.52·10-s − 2.41·11-s + 13.8·12-s + 0.832·13-s + 3.20·14-s + 4.13·15-s − 3.75·16-s − 0.485·17-s − 33.9·18-s + 1.83·19-s + 2.68·20-s − 5.23·21-s + 6.82·22-s + 0.834·23-s + 13/5·25-s − 2.35·26-s + 23.0·27-s − 3.40·28-s + 0.371·29-s − 11.6·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(130544.\) |
| Root analytic conductor: | \(2.08802\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.999770250\) |
| \(L(\frac12)\) | \(\approx\) | \(6.999770250\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 3 | \( ( 1 - T )^{8} \) | |
| 7 | \( 1 + 3 T + 8 T^{2} + 33 T^{3} + 123 T^{4} + 33 p T^{5} + 8 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| 13 | \( 1 - 3 T + 20 T^{2} - 15 T^{3} + 231 T^{4} - 15 p T^{5} + 20 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( ( 1 - 7 T + 22 T^{2} - 37 T^{3} + 57 T^{4} - 37 p T^{5} + 22 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )( 1 + p T + 4 T^{2} - 19 T^{3} - 63 T^{4} - 19 p T^{5} + 4 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 + 4 T + 30 T^{2} + 141 T^{3} + 417 T^{4} + 141 p T^{5} + 30 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 17 | \( 1 + 2 T - 20 T^{2} + 150 T^{3} + 467 T^{4} - 2935 T^{5} + 13143 T^{6} + 3209 p T^{7} - 210377 T^{8} + 3209 p^{2} T^{9} + 13143 p^{2} T^{10} - 2935 p^{3} T^{11} + 467 p^{4} T^{12} + 150 p^{5} T^{13} - 20 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( ( 1 - 4 T + 35 T^{2} - 18 T^{3} + 455 T^{4} - 18 p T^{5} + 35 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( 1 - 4 T - 42 T^{2} + 158 T^{3} + 971 T^{4} - 2891 T^{5} - 12105 T^{6} + 34329 T^{7} + 57141 T^{8} + 34329 p T^{9} - 12105 p^{2} T^{10} - 2891 p^{3} T^{11} + 971 p^{4} T^{12} + 158 p^{5} T^{13} - 42 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 2 T - 20 T^{2} - 162 T^{3} - 79 T^{4} + 6097 T^{5} + 28467 T^{6} - 124657 T^{7} - 499445 T^{8} - 124657 p T^{9} + 28467 p^{2} T^{10} + 6097 p^{3} T^{11} - 79 p^{4} T^{12} - 162 p^{5} T^{13} - 20 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 14 T + 44 T^{2} + 2 p T^{3} + 741 T^{4} - 3461 T^{5} - 53077 T^{6} + 156063 T^{7} + 844921 T^{8} + 156063 p T^{9} - 53077 p^{2} T^{10} - 3461 p^{3} T^{11} + 741 p^{4} T^{12} + 2 p^{6} T^{13} + 44 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 6 T - 120 T^{2} - 394 T^{3} + 11541 T^{4} + 22685 T^{5} - 651857 T^{6} - 236133 T^{7} + 29622571 T^{8} - 236133 p T^{9} - 651857 p^{2} T^{10} + 22685 p^{3} T^{11} + 11541 p^{4} T^{12} - 394 p^{5} T^{13} - 120 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 12 T - 6 T^{2} + 306 T^{3} + 1865 T^{4} - 9147 T^{5} - 104817 T^{6} + 676101 T^{7} - 2276181 T^{8} + 676101 p T^{9} - 104817 p^{2} T^{10} - 9147 p^{3} T^{11} + 1865 p^{4} T^{12} + 306 p^{5} T^{13} - 6 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 138 T^{2} + 146 T^{3} + 10887 T^{4} - 13213 T^{5} - 610013 T^{6} + 270903 T^{7} + 27676345 T^{8} + 270903 p T^{9} - 610013 p^{2} T^{10} - 13213 p^{3} T^{11} + 10887 p^{4} T^{12} + 146 p^{5} T^{13} - 138 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 - 7 T - 107 T^{2} + 654 T^{3} + 8042 T^{4} - 34657 T^{5} - 424428 T^{6} + 772924 T^{7} + 19410175 T^{8} + 772924 p T^{9} - 424428 p^{2} T^{10} - 34657 p^{3} T^{11} + 8042 p^{4} T^{12} + 654 p^{5} T^{13} - 107 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + T - 65 T^{2} - 1080 T^{3} + 894 T^{4} + 62409 T^{5} + 478074 T^{6} - 2019040 T^{7} - 28883847 T^{8} - 2019040 p T^{9} + 478074 p^{2} T^{10} + 62409 p^{3} T^{11} + 894 p^{4} T^{12} - 1080 p^{5} T^{13} - 65 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 16 T + 3 T^{2} + 664 T^{3} + 4500 T^{4} - 37056 T^{5} - 430631 T^{6} + 1567444 T^{7} + 18534903 T^{8} + 1567444 p T^{9} - 430631 p^{2} T^{10} - 37056 p^{3} T^{11} + 4500 p^{4} T^{12} + 664 p^{5} T^{13} + 3 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( ( 1 - 4 T + 68 T^{2} + 492 T^{3} + 422 T^{4} + 492 p T^{5} + 68 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 67 | \( ( 1 + 19 T + 285 T^{2} + 2806 T^{3} + 25511 T^{4} + 2806 p T^{5} + 285 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 20 T + 151 T^{2} - 948 T^{3} + 5336 T^{4} + 8680 T^{5} - 357951 T^{6} + 6067808 T^{7} - 73033073 T^{8} + 6067808 p T^{9} - 357951 p^{2} T^{10} + 8680 p^{3} T^{11} + 5336 p^{4} T^{12} - 948 p^{5} T^{13} + 151 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 7 T - 176 T^{2} - 557 T^{3} + 21594 T^{4} + 15702 T^{5} - 1863345 T^{6} - 909252 T^{7} + 122212573 T^{8} - 909252 p T^{9} - 1863345 p^{2} T^{10} + 15702 p^{3} T^{11} + 21594 p^{4} T^{12} - 557 p^{5} T^{13} - 176 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 24 T + 189 T^{2} - 124 T^{3} - 6786 T^{4} + 100964 T^{5} - 891383 T^{6} - 39114 p T^{7} + 106087891 T^{8} - 39114 p^{2} T^{9} - 891383 p^{2} T^{10} + 100964 p^{3} T^{11} - 6786 p^{4} T^{12} - 124 p^{5} T^{13} + 189 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 32 T + 696 T^{2} + 9651 T^{3} + 104187 T^{4} + 9651 p T^{5} + 696 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 11 T - 131 T^{2} - 2094 T^{3} + 4778 T^{4} + 126467 T^{5} - 524910 T^{6} - 1656998 T^{7} + 99643921 T^{8} - 1656998 p T^{9} - 524910 p^{2} T^{10} + 126467 p^{3} T^{11} + 4778 p^{4} T^{12} - 2094 p^{5} T^{13} - 131 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 11 T - 187 T^{2} + 962 T^{3} + 32064 T^{4} - 25625 T^{5} - 3767656 T^{6} + 3490380 T^{7} + 292157209 T^{8} + 3490380 p T^{9} - 3767656 p^{2} T^{10} - 25625 p^{3} T^{11} + 32064 p^{4} T^{12} + 962 p^{5} T^{13} - 187 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.69974784972674400570873042323, −4.63591321203489735356873195228, −4.23394200184600898038289000857, −4.22381193854412866892528132880, −4.19086229907106238405690364969, −3.89555688997293046367479551927, −3.86264152831529323823771918790, −3.51782077659365758640795794174, −3.32366304279400069560725381998, −3.26995611477277053822464063235, −3.07973055557131128897900120006, −2.97892520154490275290260366635, −2.95074838898011513045789577194, −2.81517582259553206086815275840, −2.51512883128968707311649918591, −2.43743618548672417176240905635, −2.28231594433722869996505642743, −2.18961568760642490783627567445, −2.04720323719311631953219483995, −1.47870498315137075455100032429, −1.46013544211365450603518267506, −1.29087500721472971678750099144, −0.948887359519907235415351781112, −0.930571167881182337561271294132, −0.47035343413408428510480072864, 0.47035343413408428510480072864, 0.930571167881182337561271294132, 0.948887359519907235415351781112, 1.29087500721472971678750099144, 1.46013544211365450603518267506, 1.47870498315137075455100032429, 2.04720323719311631953219483995, 2.18961568760642490783627567445, 2.28231594433722869996505642743, 2.43743618548672417176240905635, 2.51512883128968707311649918591, 2.81517582259553206086815275840, 2.95074838898011513045789577194, 2.97892520154490275290260366635, 3.07973055557131128897900120006, 3.26995611477277053822464063235, 3.32366304279400069560725381998, 3.51782077659365758640795794174, 3.86264152831529323823771918790, 3.89555688997293046367479551927, 4.19086229907106238405690364969, 4.22381193854412866892528132880, 4.23394200184600898038289000857, 4.63591321203489735356873195228, 4.69974784972674400570873042323
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.