Dirichlet series
| L(s) = 1 | − 79·4-s + 2.74e3·7-s + 6.16e3·13-s − 4.41e3·16-s + 1.08e5·19-s − 2.48e5·25-s − 2.16e5·28-s + 1.35e5·31-s + 2.38e5·37-s − 8.84e5·43-s + 4.23e6·49-s − 4.86e5·52-s − 1.58e6·61-s + 3.66e5·64-s + 1.15e7·67-s + 1.19e7·73-s − 8.57e6·76-s + 2.78e6·79-s + 1.69e7·91-s + 4.73e7·97-s + 1.96e7·100-s + 2.14e7·103-s − 3.45e7·109-s − 1.21e7·112-s − 9.73e7·121-s − 1.06e7·124-s + 127-s + ⋯ |
| L(s) = 1 | − 0.617·4-s + 3.02·7-s + 0.777·13-s − 0.269·16-s + 3.63·19-s − 3.17·25-s − 1.86·28-s + 0.814·31-s + 0.774·37-s − 1.69·43-s + 36/7·49-s − 0.479·52-s − 0.895·61-s + 0.174·64-s + 4.70·67-s + 3.58·73-s − 2.24·76-s + 0.635·79-s + 2.35·91-s + 5.26·97-s + 1.96·100-s + 1.93·103-s − 2.55·109-s − 0.815·112-s − 4.99·121-s − 0.502·124-s + 10.9·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{24} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.47644\times 10^{14}\) |
| Root analytic conductor: | \(7.68379\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{24} \cdot 7^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(49.41028114\) |
| \(L(\frac12)\) | \(\approx\) | \(49.41028114\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 - p^{3} T )^{8} \) | |
| good | 2 | \( 1 + 79 T^{2} + 2665 p^{2} T^{4} + 6443 p^{7} T^{6} + 741133 p^{8} T^{8} + 6443 p^{21} T^{10} + 2665 p^{30} T^{12} + 79 p^{42} T^{14} + p^{56} T^{16} \) |
| 5 | \( 1 + 248224 T^{2} + 43968761854 T^{4} + 203987509658488 p^{2} T^{6} + 745629947608940359 p^{4} T^{8} + 203987509658488 p^{16} T^{10} + 43968761854 p^{28} T^{12} + 248224 p^{42} T^{14} + p^{56} T^{16} \) | |
| 11 | \( 1 + 97300816 T^{2} + 4404266796931006 T^{4} + \)\(12\!\cdots\!32\)\( T^{6} + \)\(27\!\cdots\!67\)\( T^{8} + \)\(12\!\cdots\!32\)\( p^{14} T^{10} + 4404266796931006 p^{28} T^{12} + 97300816 p^{42} T^{14} + p^{56} T^{16} \) | |
| 13 | \( ( 1 - 3080 T + 40179784 T^{2} + 66994044856 T^{3} + 5023582868166430 T^{4} + 66994044856 p^{7} T^{5} + 40179784 p^{14} T^{6} - 3080 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 17 | \( 1 + 1297665064 T^{2} + 1000029782281331836 T^{4} + \)\(59\!\cdots\!20\)\( T^{6} + \)\(28\!\cdots\!26\)\( T^{8} + \)\(59\!\cdots\!20\)\( p^{14} T^{10} + 1000029782281331836 p^{28} T^{12} + 1297665064 p^{42} T^{14} + p^{56} T^{16} \) | |
| 19 | \( ( 1 - 54296 T + 1766210014 T^{2} - 38954064060776 T^{3} + 599754509764421191 T^{4} - 38954064060776 p^{7} T^{5} + 1766210014 p^{14} T^{6} - 54296 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 23 | \( 1 + 15439416232 T^{2} + \)\(11\!\cdots\!82\)\( T^{4} + \)\(59\!\cdots\!64\)\( T^{6} + \)\(22\!\cdots\!39\)\( T^{8} + \)\(59\!\cdots\!64\)\( p^{14} T^{10} + \)\(11\!\cdots\!82\)\( p^{28} T^{12} + 15439416232 p^{42} T^{14} + p^{56} T^{16} \) | |
| 29 | \( 1 + 118589358544 T^{2} + \)\(63\!\cdots\!96\)\( T^{4} + \)\(70\!\cdots\!12\)\( p T^{6} + \)\(43\!\cdots\!42\)\( T^{8} + \)\(70\!\cdots\!12\)\( p^{15} T^{10} + \)\(63\!\cdots\!96\)\( p^{28} T^{12} + 118589358544 p^{42} T^{14} + p^{56} T^{16} \) | |
| 31 | \( ( 1 - 67520 T + 66595331002 T^{2} - 5735267839426304 T^{3} + \)\(24\!\cdots\!27\)\( T^{4} - 5735267839426304 p^{7} T^{5} + 66595331002 p^{14} T^{6} - 67520 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 37 | \( ( 1 - 119348 T + 189262852894 T^{2} - 23159482504157432 T^{3} + \)\(23\!\cdots\!31\)\( T^{4} - 23159482504157432 p^{7} T^{5} + 189262852894 p^{14} T^{6} - 119348 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 41 | \( 1 + 1085497769416 T^{2} + \)\(58\!\cdots\!22\)\( T^{4} + \)\(19\!\cdots\!40\)\( T^{6} + \)\(45\!\cdots\!91\)\( T^{8} + \)\(19\!\cdots\!40\)\( p^{14} T^{10} + \)\(58\!\cdots\!22\)\( p^{28} T^{12} + 1085497769416 p^{42} T^{14} + p^{56} T^{16} \) | |
| 43 | \( ( 1 + 442120 T + 702197112472 T^{2} + 349093335248830312 T^{3} + \)\(25\!\cdots\!02\)\( T^{4} + 349093335248830312 p^{7} T^{5} + 702197112472 p^{14} T^{6} + 442120 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 47 | \( 1 + 1233710781280 T^{2} + \)\(11\!\cdots\!68\)\( T^{4} + \)\(79\!\cdots\!52\)\( T^{6} + \)\(47\!\cdots\!42\)\( T^{8} + \)\(79\!\cdots\!52\)\( p^{14} T^{10} + \)\(11\!\cdots\!68\)\( p^{28} T^{12} + 1233710781280 p^{42} T^{14} + p^{56} T^{16} \) | |
| 53 | \( 1 + 3804140900872 T^{2} + \)\(75\!\cdots\!52\)\( T^{4} + \)\(12\!\cdots\!44\)\( T^{6} + \)\(17\!\cdots\!54\)\( T^{8} + \)\(12\!\cdots\!44\)\( p^{14} T^{10} + \)\(75\!\cdots\!52\)\( p^{28} T^{12} + 3804140900872 p^{42} T^{14} + p^{56} T^{16} \) | |
| 59 | \( 1 + 8867691232096 T^{2} + \)\(43\!\cdots\!32\)\( T^{4} + \)\(15\!\cdots\!40\)\( T^{6} + \)\(43\!\cdots\!06\)\( T^{8} + \)\(15\!\cdots\!40\)\( p^{14} T^{10} + \)\(43\!\cdots\!32\)\( p^{28} T^{12} + 8867691232096 p^{42} T^{14} + p^{56} T^{16} \) | |
| 61 | \( ( 1 + 793336 T + 104189206564 p T^{2} - 3045590906131039736 T^{3} + \)\(16\!\cdots\!10\)\( T^{4} - 3045590906131039736 p^{7} T^{5} + 104189206564 p^{15} T^{6} + 793336 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 67 | \( ( 1 - 5787920 T + 26031974029024 T^{2} - 84563244830441223488 T^{3} + \)\(24\!\cdots\!46\)\( T^{4} - 84563244830441223488 p^{7} T^{5} + 26031974029024 p^{14} T^{6} - 5787920 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 71 | \( 1 + 47681888788864 T^{2} + \)\(11\!\cdots\!82\)\( T^{4} + \)\(17\!\cdots\!80\)\( T^{6} + \)\(18\!\cdots\!71\)\( T^{8} + \)\(17\!\cdots\!80\)\( p^{14} T^{10} + \)\(11\!\cdots\!82\)\( p^{28} T^{12} + 47681888788864 p^{42} T^{14} + p^{56} T^{16} \) | |
| 73 | \( ( 1 - 5962544 T + 42592384764640 T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(60\!\cdots\!54\)\( T^{4} - \)\(13\!\cdots\!00\)\( p^{7} T^{5} + 42592384764640 p^{14} T^{6} - 5962544 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 79 | \( ( 1 - 1392152 T + 35016092163592 T^{2} + 70854268528242774376 T^{3} + \)\(47\!\cdots\!34\)\( T^{4} + 70854268528242774376 p^{7} T^{5} + 35016092163592 p^{14} T^{6} - 1392152 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 83 | \( 1 + 63867874982848 T^{2} + \)\(29\!\cdots\!92\)\( T^{4} + \)\(80\!\cdots\!76\)\( T^{6} + \)\(23\!\cdots\!54\)\( T^{8} + \)\(80\!\cdots\!76\)\( p^{14} T^{10} + \)\(29\!\cdots\!92\)\( p^{28} T^{12} + 63867874982848 p^{42} T^{14} + p^{56} T^{16} \) | |
| 89 | \( 1 + 195951162799144 T^{2} + \)\(18\!\cdots\!26\)\( T^{4} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(56\!\cdots\!07\)\( T^{8} + \)\(11\!\cdots\!68\)\( p^{14} T^{10} + \)\(18\!\cdots\!26\)\( p^{28} T^{12} + 195951162799144 p^{42} T^{14} + p^{56} T^{16} \) | |
| 97 | \( ( 1 - 23676368 T + 474059728341124 T^{2} - \)\(57\!\cdots\!00\)\( T^{3} + \)\(61\!\cdots\!42\)\( T^{4} - \)\(57\!\cdots\!00\)\( p^{7} T^{5} + 474059728341124 p^{14} T^{6} - 23676368 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
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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
−4.52789249841865795679245875086, −3.93190556290811572495790921478, −3.79926158822268745556370839376, −3.79294001509310178139639461326, −3.76114982661074989731846803544, −3.74886063411338157035410528161, −3.72520570929848388310513019567, −3.10433660202014612294048863582, −3.08454066618850905165408863869, −2.81868994199544454926970015459, −2.51083491104148447980795661343, −2.49739024785556185809759434760, −2.44118473299472025170835762488, −1.89809031229172624645743852055, −1.83283858394141984133162179883, −1.83125128460108259127330347237, −1.65535894310068822158180881201, −1.58528147889166317621469793902, −1.09087433707234381890308902946, −1.02571808795740977658712822575, −0.914510256649863191316090241706, −0.798817280392591657192065750894, −0.49753099829598004473579984787, −0.40982810569794526981662476957, −0.30400135833887529278886288721, 0.30400135833887529278886288721, 0.40982810569794526981662476957, 0.49753099829598004473579984787, 0.798817280392591657192065750894, 0.914510256649863191316090241706, 1.02571808795740977658712822575, 1.09087433707234381890308902946, 1.58528147889166317621469793902, 1.65535894310068822158180881201, 1.83125128460108259127330347237, 1.83283858394141984133162179883, 1.89809031229172624645743852055, 2.44118473299472025170835762488, 2.49739024785556185809759434760, 2.51083491104148447980795661343, 2.81868994199544454926970015459, 3.08454066618850905165408863869, 3.10433660202014612294048863582, 3.72520570929848388310513019567, 3.74886063411338157035410528161, 3.76114982661074989731846803544, 3.79294001509310178139639461326, 3.79926158822268745556370839376, 3.93190556290811572495790921478, 4.52789249841865795679245875086
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.