Dirichlet series
| L(s) = 1 | − 10·2-s − 33·4-s + 1.02e3·8-s − 2.14e3·11-s − 1.14e4·16-s + 2.14e4·22-s − 3.04e4·23-s + 4.02e4·25-s − 3.25e4·29-s + 6.37e4·32-s + 9.13e4·37-s − 4.45e5·43-s + 7.06e4·44-s + 3.04e5·46-s − 4.02e5·50-s − 2.60e4·53-s + 3.25e5·58-s + 2.24e5·64-s − 7.68e5·67-s − 2.25e5·71-s − 9.13e5·74-s + 1.11e6·79-s + 4.45e6·86-s − 2.19e6·88-s + 1.00e6·92-s − 1.32e6·100-s + 2.60e5·106-s + ⋯ |
| L(s) = 1 | − 5/4·2-s − 0.515·4-s + 2.00·8-s − 1.60·11-s − 2.78·16-s + 2.00·22-s − 2.50·23-s + 2.57·25-s − 1.33·29-s + 1.94·32-s + 1.80·37-s − 5.60·43-s + 0.829·44-s + 3.12·46-s − 3.21·50-s − 0.175·53-s + 1.66·58-s + 0.858·64-s − 2.55·67-s − 0.630·71-s − 2.25·74-s + 2.26·79-s + 7.00·86-s − 3.22·88-s + 1.29·92-s − 1.32·100-s + 0.218·106-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.12240\times 10^{16}\) |
| Root analytic conductor: | \(10.0724\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 7^{16} ,\ ( \ : [3]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(0.002337821503\) |
| \(L(\frac12)\) | \(\approx\) | \(0.002337821503\) |
| \(L(4)\) | 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 | \( 1 \) | |
| good | 2 | \( ( 1 + 5 T + 27 p T^{2} + 23 p T^{3} + 265 p^{4} T^{4} + 23 p^{7} T^{5} + 27 p^{13} T^{6} + 5 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 5 | \( 1 - 40226 T^{2} + 1130825449 T^{4} - 945309374954 p^{2} T^{6} + 685993727134996 p^{4} T^{8} - 945309374954 p^{14} T^{10} + 1130825449 p^{24} T^{12} - 40226 p^{36} T^{14} + p^{48} T^{16} \) | |
| 11 | \( ( 1 + 1070 T + 6822033 T^{2} + 5373825550 T^{3} + 17953528514164 T^{4} + 5373825550 p^{6} T^{5} + 6822033 p^{12} T^{6} + 1070 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 13 | \( 1 - 16298786 T^{2} + 111852524426401 T^{4} - 37539779661680550026 p T^{6} + \)\(20\!\cdots\!88\)\( T^{8} - 37539779661680550026 p^{13} T^{10} + 111852524426401 p^{24} T^{12} - 16298786 p^{36} T^{14} + p^{48} T^{16} \) | |
| 17 | \( 1 - 23889824 T^{2} + 122207207828156 p T^{4} - \)\(38\!\cdots\!92\)\( T^{6} + \)\(17\!\cdots\!42\)\( T^{8} - \)\(38\!\cdots\!92\)\( p^{12} T^{10} + 122207207828156 p^{25} T^{12} - 23889824 p^{36} T^{14} + p^{48} T^{16} \) | |
| 19 | \( 1 - 120011066 T^{2} + 4558960827908065 T^{4} + \)\(74\!\cdots\!14\)\( p T^{6} - \)\(16\!\cdots\!56\)\( T^{8} + \)\(74\!\cdots\!14\)\( p^{13} T^{10} + 4558960827908065 p^{24} T^{12} - 120011066 p^{36} T^{14} + p^{48} T^{16} \) | |
| 23 | \( ( 1 + 15224 T + 303174876 T^{2} + 3933308321128 T^{3} + 71062389247076086 T^{4} + 3933308321128 p^{6} T^{5} + 303174876 p^{12} T^{6} + 15224 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 29 | \( ( 1 + 16262 T + 1936712745 T^{2} + 19588912728706 T^{3} + 1557131395176891808 T^{4} + 19588912728706 p^{6} T^{5} + 1936712745 p^{12} T^{6} + 16262 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 31 | \( 1 - 3071909540 T^{2} + 5609111825159464114 T^{4} - \)\(68\!\cdots\!20\)\( T^{6} + \)\(68\!\cdots\!91\)\( T^{8} - \)\(68\!\cdots\!20\)\( p^{12} T^{10} + 5609111825159464114 p^{24} T^{12} - 3071909540 p^{36} T^{14} + p^{48} T^{16} \) | |
| 37 | \( ( 1 - 45670 T + 7173593881 T^{2} - 327580206713990 T^{3} + 24148714940087622596 T^{4} - 327580206713990 p^{6} T^{5} + 7173593881 p^{12} T^{6} - 45670 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 41 | \( 1 - 26005847576 T^{2} + \)\(34\!\cdots\!60\)\( T^{4} - \)\(28\!\cdots\!64\)\( T^{6} + \)\(15\!\cdots\!14\)\( T^{8} - \)\(28\!\cdots\!64\)\( p^{12} T^{10} + \)\(34\!\cdots\!60\)\( p^{24} T^{12} - 26005847576 p^{36} T^{14} + p^{48} T^{16} \) | |
| 43 | \( ( 1 + 222830 T + 23708395369 T^{2} + 1380667882433470 T^{3} + 73296626267097035876 T^{4} + 1380667882433470 p^{6} T^{5} + 23708395369 p^{12} T^{6} + 222830 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 47 | \( 1 - 59192143520 T^{2} + \)\(16\!\cdots\!20\)\( T^{4} - \)\(27\!\cdots\!64\)\( T^{6} + \)\(34\!\cdots\!70\)\( T^{8} - \)\(27\!\cdots\!64\)\( p^{12} T^{10} + \)\(16\!\cdots\!20\)\( p^{24} T^{12} - 59192143520 p^{36} T^{14} + p^{48} T^{16} \) | |
| 53 | \( ( 1 + 13034 T + 38080896873 T^{2} + 63769766876950 T^{3} + \)\(11\!\cdots\!52\)\( T^{4} + 63769766876950 p^{6} T^{5} + 38080896873 p^{12} T^{6} + 13034 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 59 | \( 1 - 34928672738 T^{2} + \)\(31\!\cdots\!25\)\( T^{4} - \)\(27\!\cdots\!74\)\( T^{6} + \)\(40\!\cdots\!28\)\( T^{8} - \)\(27\!\cdots\!74\)\( p^{12} T^{10} + \)\(31\!\cdots\!25\)\( p^{24} T^{12} - 34928672738 p^{36} T^{14} + p^{48} T^{16} \) | |
| 61 | \( 1 - 353197951976 T^{2} + \)\(57\!\cdots\!92\)\( T^{4} - \)\(55\!\cdots\!08\)\( T^{6} + \)\(34\!\cdots\!42\)\( T^{8} - \)\(55\!\cdots\!08\)\( p^{12} T^{10} + \)\(57\!\cdots\!92\)\( p^{24} T^{12} - 353197951976 p^{36} T^{14} + p^{48} T^{16} \) | |
| 67 | \( ( 1 + 384094 T + 316936300861 T^{2} + 89519136577479610 T^{3} + \)\(41\!\cdots\!76\)\( T^{4} + 89519136577479610 p^{6} T^{5} + 316936300861 p^{12} T^{6} + 384094 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 71 | \( ( 1 + 112844 T + 324448495296 T^{2} - 1073384954931548 T^{3} + \)\(47\!\cdots\!38\)\( T^{4} - 1073384954931548 p^{6} T^{5} + 324448495296 p^{12} T^{6} + 112844 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 73 | \( 1 - 979051001426 T^{2} + \)\(44\!\cdots\!89\)\( T^{4} - \)\(12\!\cdots\!38\)\( T^{6} + \)\(22\!\cdots\!56\)\( T^{8} - \)\(12\!\cdots\!38\)\( p^{12} T^{10} + \)\(44\!\cdots\!89\)\( p^{24} T^{12} - 979051001426 p^{36} T^{14} + p^{48} T^{16} \) | |
| 79 | \( ( 1 - 559592 T + 506004632830 T^{2} - 161020277829667928 T^{3} + \)\(12\!\cdots\!43\)\( T^{4} - 161020277829667928 p^{6} T^{5} + 506004632830 p^{12} T^{6} - 559592 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 83 | \( 1 - 577456669706 T^{2} + \)\(27\!\cdots\!89\)\( T^{4} - \)\(12\!\cdots\!18\)\( T^{6} + \)\(42\!\cdots\!16\)\( T^{8} - \)\(12\!\cdots\!18\)\( p^{12} T^{10} + \)\(27\!\cdots\!89\)\( p^{24} T^{12} - 577456669706 p^{36} T^{14} + p^{48} T^{16} \) | |
| 89 | \( 1 - 3692886607664 T^{2} + \)\(60\!\cdots\!64\)\( T^{4} - \)\(58\!\cdots\!12\)\( T^{6} + \)\(36\!\cdots\!06\)\( T^{8} - \)\(58\!\cdots\!12\)\( p^{12} T^{10} + \)\(60\!\cdots\!64\)\( p^{24} T^{12} - 3692886607664 p^{36} T^{14} + p^{48} T^{16} \) | |
| 97 | \( 1 - 4315064900690 T^{2} + \)\(91\!\cdots\!41\)\( T^{4} - \)\(12\!\cdots\!58\)\( T^{6} + \)\(12\!\cdots\!60\)\( T^{8} - \)\(12\!\cdots\!58\)\( p^{12} T^{10} + \)\(91\!\cdots\!41\)\( p^{24} T^{12} - 4315064900690 p^{36} T^{14} + p^{48} T^{16} \) | |
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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
−3.85653135224836531059358550750, −3.72607130372666437187579326886, −3.67751393747086828393537507630, −3.64128844788389571492449407881, −3.07236362080416272995893717224, −2.95471828625692777197821580448, −2.93508333677536815954268197293, −2.92134824106961500338089522162, −2.61609326933337475897445956593, −2.48065595633735548272040943308, −2.45409835118049573935367522791, −2.04222945828047954588374468268, −2.00175247726547330687769730926, −1.91533606182803200528924018756, −1.73991201722493053836979266337, −1.52388816523098548411892061100, −1.37891429255046378019131537851, −1.21521547605773007440159677235, −1.02393295537140872069217436584, −0.871647903772042632494551239891, −0.861038837306475752056865618136, −0.30450378387122134621053266001, −0.26219142950760228094772109147, −0.18695980802146359532578176909, −0.01556310836664396114042772613, 0.01556310836664396114042772613, 0.18695980802146359532578176909, 0.26219142950760228094772109147, 0.30450378387122134621053266001, 0.861038837306475752056865618136, 0.871647903772042632494551239891, 1.02393295537140872069217436584, 1.21521547605773007440159677235, 1.37891429255046378019131537851, 1.52388816523098548411892061100, 1.73991201722493053836979266337, 1.91533606182803200528924018756, 2.00175247726547330687769730926, 2.04222945828047954588374468268, 2.45409835118049573935367522791, 2.48065595633735548272040943308, 2.61609326933337475897445956593, 2.92134824106961500338089522162, 2.93508333677536815954268197293, 2.95471828625692777197821580448, 3.07236362080416272995893717224, 3.64128844788389571492449407881, 3.67751393747086828393537507630, 3.72607130372666437187579326886, 3.85653135224836531059358550750
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.