Dirichlet series
| L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 4·6-s − 2·7-s + 9-s + 8·11-s − 2·12-s + 6·13-s − 4·14-s + 2·18-s − 4·19-s + 4·21-s + 16·22-s + 13·25-s + 12·26-s − 2·28-s + 2·29-s − 4·31-s − 2·32-s − 16·33-s + 36-s − 24·37-s − 8·38-s − 12·39-s + 8·42-s + 8·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.63·6-s − 0.755·7-s + 1/3·9-s + 2.41·11-s − 0.577·12-s + 1.66·13-s − 1.06·14-s + 0.471·18-s − 0.917·19-s + 0.872·21-s + 3.41·22-s + 13/5·25-s + 2.35·26-s − 0.377·28-s + 0.371·29-s − 0.718·31-s − 0.353·32-s − 2.78·33-s + 1/6·36-s − 3.94·37-s − 1.29·38-s − 1.92·39-s + 1.23·42-s + 1.21·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(34304.6\) |
| Root analytic conductor: | \(1.92070\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.7133041527\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7133041527\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 7 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 - 8 T + 13 T^{2} + 74 T^{3} - 435 T^{4} + 74 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( 1 - 13 T^{2} + 39 T^{4} + 313 T^{6} - 3064 T^{8} + 313 p^{2} T^{10} + 39 p^{4} T^{12} - 13 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 6 T + 16 T^{2} - 54 T^{3} + 303 T^{4} - 1626 T^{5} + 5204 T^{6} - 13212 T^{7} + 50813 T^{8} - 13212 p T^{9} + 5204 p^{2} T^{10} - 1626 p^{3} T^{11} + 303 p^{4} T^{12} - 54 p^{5} T^{13} + 16 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 2 T^{2} - 105 T^{4} - 1680 T^{5} + 4268 T^{6} + 19080 T^{7} + 19229 T^{8} + 19080 p T^{9} + 4268 p^{2} T^{10} - 1680 p^{3} T^{11} - 105 p^{4} T^{12} - 2 p^{6} T^{14} + p^{8} T^{16} \) | |
| 19 | \( 1 + 4 T + T^{2} - 70 T^{3} - 577 T^{4} - 2630 T^{5} + 1679 T^{6} + 43964 T^{7} + 183568 T^{8} + 43964 p T^{9} + 1679 p^{2} T^{10} - 2630 p^{3} T^{11} - 577 p^{4} T^{12} - 70 p^{5} T^{13} + p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 61 T^{2} - 90 T^{3} + 1677 T^{4} - 90 p T^{5} + 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 - 2 T - 40 T^{2} + 360 T^{3} - 185 T^{4} - 17346 T^{5} + 75842 T^{6} + 243760 T^{7} - 3491915 T^{8} + 243760 p T^{9} + 75842 p^{2} T^{10} - 17346 p^{3} T^{11} - 185 p^{4} T^{12} + 360 p^{5} T^{13} - 40 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 4 T - 23 T^{2} - 94 T^{3} - 49 T^{4} - 3230 T^{5} + 15983 T^{6} + 24260 T^{7} - 841088 T^{8} + 24260 p T^{9} + 15983 p^{2} T^{10} - 3230 p^{3} T^{11} - 49 p^{4} T^{12} - 94 p^{5} T^{13} - 23 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 24 T + 306 T^{2} + 2552 T^{3} + 14631 T^{4} + 44312 T^{5} - 176660 T^{6} - 3619200 T^{7} - 28521323 T^{8} - 3619200 p T^{9} - 176660 p^{2} T^{10} + 44312 p^{3} T^{11} + 14631 p^{4} T^{12} + 2552 p^{5} T^{13} + 306 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 50 T^{2} + 240 T^{3} + 2199 T^{4} - 7920 T^{5} - 33700 T^{6} + 212520 T^{7} + 2579021 T^{8} + 212520 p T^{9} - 33700 p^{2} T^{10} - 7920 p^{3} T^{11} + 2199 p^{4} T^{12} + 240 p^{5} T^{13} - 50 p^{6} T^{14} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 4 T + 132 T^{2} - 428 T^{3} + 7898 T^{4} - 428 p T^{5} + 132 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 2 T - 24 T^{2} - 300 T^{3} - 2809 T^{4} - 20694 T^{5} + 1670 T^{6} + 1342496 T^{7} + 8656437 T^{8} + 1342496 p T^{9} + 1670 p^{2} T^{10} - 20694 p^{3} T^{11} - 2809 p^{4} T^{12} - 300 p^{5} T^{13} - 24 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 22 T + 180 T^{2} + 888 T^{3} + 7199 T^{4} + 96966 T^{5} + 1007894 T^{6} + 6420736 T^{7} + 35667381 T^{8} + 6420736 p T^{9} + 1007894 p^{2} T^{10} + 96966 p^{3} T^{11} + 7199 p^{4} T^{12} + 888 p^{5} T^{13} + 180 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 10 T + 20 T^{2} - 60 T^{3} + 3799 T^{4} - 42450 T^{5} + 117290 T^{6} - 367120 T^{7} + 8698021 T^{8} - 367120 p T^{9} + 117290 p^{2} T^{10} - 42450 p^{3} T^{11} + 3799 p^{4} T^{12} - 60 p^{5} T^{13} + 20 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 26 T + 340 T^{2} - 3640 T^{3} + 34415 T^{4} - 294698 T^{5} + 44678 p T^{6} - 24412480 T^{7} + 195676405 T^{8} - 24412480 p T^{9} + 44678 p^{3} T^{10} - 294698 p^{3} T^{11} + 34415 p^{4} T^{12} - 3640 p^{5} T^{13} + 340 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 - 2 T + 126 T^{2} - 574 T^{3} + 10502 T^{4} - 574 p T^{5} + 126 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 16 T + 110 T^{2} + 360 T^{3} - 2765 T^{4} - 56592 T^{5} - 98932 T^{6} + 2943520 T^{7} + 35567845 T^{8} + 2943520 p T^{9} - 98932 p^{2} T^{10} - 56592 p^{3} T^{11} - 2765 p^{4} T^{12} + 360 p^{5} T^{13} + 110 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 2 T + 4 T^{2} + 304 T^{3} - 4489 T^{4} - 57454 T^{5} + 69230 T^{6} + 2015200 T^{7} - 11723483 T^{8} + 2015200 p T^{9} + 69230 p^{2} T^{10} - 57454 p^{3} T^{11} - 4489 p^{4} T^{12} + 304 p^{5} T^{13} + 4 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 12 T - 62 T^{2} - 438 T^{3} + 7131 T^{4} + 37380 T^{5} + 741842 T^{6} + 2884140 T^{7} - 60254563 T^{8} + 2884140 p T^{9} + 741842 p^{2} T^{10} + 37380 p^{3} T^{11} + 7131 p^{4} T^{12} - 438 p^{5} T^{13} - 62 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 38 T + 600 T^{2} + 6132 T^{3} + 61679 T^{4} + 542814 T^{5} + 3329414 T^{6} + 28696544 T^{7} + 326181861 T^{8} + 28696544 p T^{9} + 3329414 p^{2} T^{10} + 542814 p^{3} T^{11} + 61679 p^{4} T^{12} + 6132 p^{5} T^{13} + 600 p^{6} T^{14} + 38 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 6 T + 271 T^{2} + 1620 T^{3} + 32625 T^{4} + 1620 p T^{5} + 271 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 6 T + 48 T^{2} - 1132 T^{3} + 15111 T^{4} - 102286 T^{5} + 1142410 T^{6} - 17440416 T^{7} + 145187749 T^{8} - 17440416 p T^{9} + 1142410 p^{2} T^{10} - 102286 p^{3} T^{11} + 15111 p^{4} T^{12} - 1132 p^{5} T^{13} + 48 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.78630795755661295906857298250, −4.68291846048525069387313727089, −4.67626630969032733324135923216, −4.55504068662840320934763850438, −4.13696369672813120520699858455, −4.07990367737332918872323447510, −4.04137956229547846298880715261, −3.96342461850047429372857781065, −3.88307925096935197776707105994, −3.69773011825359036172571449269, −3.37993851486616771145193616861, −3.21783210837416938015018182641, −3.15761926783736901780030235946, −3.10025752310759595141525478417, −2.85492963155460446983182374479, −2.79417839993772989984011628802, −2.39997074364338936707924314890, −1.95388914810413078260587428648, −1.94495867303911310367797456016, −1.77876089155890585001076755936, −1.62523047471475921994580916532, −1.15386876949962160614081465805, −0.999587154574059459307954624636, −0.993385182787748045894151527706, −0.12765300321659256116737686567, 0.12765300321659256116737686567, 0.993385182787748045894151527706, 0.999587154574059459307954624636, 1.15386876949962160614081465805, 1.62523047471475921994580916532, 1.77876089155890585001076755936, 1.94495867303911310367797456016, 1.95388914810413078260587428648, 2.39997074364338936707924314890, 2.79417839993772989984011628802, 2.85492963155460446983182374479, 3.10025752310759595141525478417, 3.15761926783736901780030235946, 3.21783210837416938015018182641, 3.37993851486616771145193616861, 3.69773011825359036172571449269, 3.88307925096935197776707105994, 3.96342461850047429372857781065, 4.04137956229547846298880715261, 4.07990367737332918872323447510, 4.13696369672813120520699858455, 4.55504068662840320934763850438, 4.67626630969032733324135923216, 4.68291846048525069387313727089, 4.78630795755661295906857298250
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.