Dirichlet series
| L(s) = 1 | + 8·2-s + 32·4-s + 6·5-s + 2·7-s + 80·8-s + 12·9-s + 48·10-s + 12·11-s + 16·14-s + 120·16-s + 96·18-s + 44·19-s + 192·20-s + 96·22-s + 18·25-s + 64·28-s − 72·29-s + 94·31-s + 32·32-s + 12·35-s + 384·36-s + 46·37-s + 352·38-s + 480·40-s + 30·41-s + 384·44-s + 72·45-s + ⋯ |
| L(s) = 1 | + 4·2-s + 8·4-s + 6/5·5-s + 2/7·7-s + 10·8-s + 4/3·9-s + 24/5·10-s + 1.09·11-s + 8/7·14-s + 15/2·16-s + 16/3·18-s + 2.31·19-s + 48/5·20-s + 4.36·22-s + 0.719·25-s + 16/7·28-s − 2.48·29-s + 3.03·31-s + 32-s + 0.342·35-s + 32/3·36-s + 1.24·37-s + 9.26·38-s + 12·40-s + 0.731·41-s + 8.72·44-s + 8/5·45-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.39612\times 10^{11}\) |
| Root analytic conductor: | \(5.25637\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 13^{16} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(321.9916756\) |
| \(L(\frac12)\) | \(\approx\) | \(321.9916756\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p T^{2} )^{4} \) |
| 3 | \( ( 1 - p T^{2} )^{4} \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 6 T + 18 T^{2} + 132 T^{3} - 547 p T^{4} + 11484 T^{5} - 10962 T^{6} - 279258 T^{7} + 2657664 T^{8} - 279258 p^{2} T^{9} - 10962 p^{4} T^{10} + 11484 p^{6} T^{11} - 547 p^{9} T^{12} + 132 p^{10} T^{13} + 18 p^{12} T^{14} - 6 p^{14} T^{15} + p^{16} T^{16} \) |
| 7 | \( 1 - 2 T + 2 T^{2} + 8 p T^{3} - 4159 T^{4} + 2444 T^{5} + 102 p^{2} T^{6} - 31062 p T^{7} + 10452640 T^{8} - 31062 p^{3} T^{9} + 102 p^{6} T^{10} + 2444 p^{6} T^{11} - 4159 p^{8} T^{12} + 8 p^{11} T^{13} + 2 p^{12} T^{14} - 2 p^{14} T^{15} + p^{16} T^{16} \) | |
| 11 | \( 1 - 12 T + 72 T^{2} + 420 T^{3} - 38684 T^{4} + 327276 T^{5} - 1053864 T^{6} - 24345252 T^{7} + 740847750 T^{8} - 24345252 p^{2} T^{9} - 1053864 p^{4} T^{10} + 327276 p^{6} T^{11} - 38684 p^{8} T^{12} + 420 p^{10} T^{13} + 72 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 - 1070 T^{2} + 712633 T^{4} - 317297750 T^{6} + 106628937748 T^{8} - 317297750 p^{4} T^{10} + 712633 p^{8} T^{12} - 1070 p^{12} T^{14} + p^{16} T^{16} \) | |
| 19 | \( 1 - 44 T + 968 T^{2} - 16852 T^{3} + 21968 p T^{4} - 11599852 T^{5} + 248352984 T^{6} - 4235164692 T^{7} + 72205037854 T^{8} - 4235164692 p^{2} T^{9} + 248352984 p^{4} T^{10} - 11599852 p^{6} T^{11} + 21968 p^{9} T^{12} - 16852 p^{10} T^{13} + 968 p^{12} T^{14} - 44 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 - 64 p T^{2} + 1016764 T^{4} - 462643520 T^{6} + 217439885062 T^{8} - 462643520 p^{4} T^{10} + 1016764 p^{8} T^{12} - 64 p^{13} T^{14} + p^{16} T^{16} \) | |
| 29 | \( ( 1 + 36 T + 2377 T^{2} + 61620 T^{3} + 2804004 T^{4} + 61620 p^{2} T^{5} + 2377 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 31 | \( 1 - 94 T + 4418 T^{2} - 169400 T^{3} + 6442529 T^{4} - 230690348 T^{5} + 7877190 p^{2} T^{6} - 7537605690 p T^{7} + 7138777033984 T^{8} - 7537605690 p^{3} T^{9} + 7877190 p^{6} T^{10} - 230690348 p^{6} T^{11} + 6442529 p^{8} T^{12} - 169400 p^{10} T^{13} + 4418 p^{12} T^{14} - 94 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 - 46 T + 1058 T^{2} - 11108 T^{3} + 24053 T^{4} + 34279876 T^{5} - 1540628538 T^{6} + 127996700718 T^{7} - 5642962167932 T^{8} + 127996700718 p^{2} T^{9} - 1540628538 p^{4} T^{10} + 34279876 p^{6} T^{11} + 24053 p^{8} T^{12} - 11108 p^{10} T^{13} + 1058 p^{12} T^{14} - 46 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 - 30 T + 450 T^{2} - 65520 T^{3} - 1066247 T^{4} + 3164220 p T^{5} - 1265744250 T^{6} + 116120560170 T^{7} - 6270568853904 T^{8} + 116120560170 p^{2} T^{9} - 1265744250 p^{4} T^{10} + 3164220 p^{7} T^{11} - 1066247 p^{8} T^{12} - 65520 p^{10} T^{13} + 450 p^{12} T^{14} - 30 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 - 7634 T^{2} + 32581105 T^{4} - 94619715410 T^{6} + 202084446051076 T^{8} - 94619715410 p^{4} T^{10} + 32581105 p^{8} T^{12} - 7634 p^{12} T^{14} + p^{16} T^{16} \) | |
| 47 | \( 1 + 300 T + 45000 T^{2} + 4678428 T^{3} + 389267620 T^{4} + 27657860724 T^{5} + 1724159592792 T^{6} + 95617768295268 T^{7} + 4746944876916294 T^{8} + 95617768295268 p^{2} T^{9} + 1724159592792 p^{4} T^{10} + 27657860724 p^{6} T^{11} + 389267620 p^{8} T^{12} + 4678428 p^{10} T^{13} + 45000 p^{12} T^{14} + 300 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( ( 1 - 42 T + 9025 T^{2} - 319362 T^{3} + 34812360 T^{4} - 319362 p^{2} T^{5} + 9025 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 59 | \( 1 - 12 T + 72 T^{2} + 75108 T^{3} - 4867772 T^{4} - 454056660 T^{5} + 8619765336 T^{6} - 1038798190884 T^{7} + 82841566547526 T^{8} - 1038798190884 p^{2} T^{9} + 8619765336 p^{4} T^{10} - 454056660 p^{6} T^{11} - 4867772 p^{8} T^{12} + 75108 p^{10} T^{13} + 72 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( ( 1 - 90 T + 9202 T^{2} - 263472 T^{3} + 23608959 T^{4} - 263472 p^{2} T^{5} + 9202 p^{4} T^{6} - 90 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 67 | \( 1 - 74 T + 2738 T^{2} - 31600 T^{3} - 2130847 T^{4} - 1176894916 T^{5} + 93423762870 T^{6} - 7120448267226 T^{7} + 584126519344768 T^{8} - 7120448267226 p^{2} T^{9} + 93423762870 p^{4} T^{10} - 1176894916 p^{6} T^{11} - 2130847 p^{8} T^{12} - 31600 p^{10} T^{13} + 2738 p^{12} T^{14} - 74 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( 1 + 156 T + 12168 T^{2} + 766188 T^{3} + 72759268 T^{4} + 7862233572 T^{5} + 634695689880 T^{6} + 42874795546548 T^{7} + 2889990280855878 T^{8} + 42874795546548 p^{2} T^{9} + 634695689880 p^{4} T^{10} + 7862233572 p^{6} T^{11} + 72759268 p^{8} T^{12} + 766188 p^{10} T^{13} + 12168 p^{12} T^{14} + 156 p^{14} T^{15} + p^{16} T^{16} \) | |
| 73 | \( 1 + 16 T + 128 T^{2} - 140416 T^{3} + 32181122 T^{4} + 2514741392 T^{5} + 45975005184 T^{6} + 6743944510608 T^{7} + 38152958606083 T^{8} + 6743944510608 p^{2} T^{9} + 45975005184 p^{4} T^{10} + 2514741392 p^{6} T^{11} + 32181122 p^{8} T^{12} - 140416 p^{10} T^{13} + 128 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( ( 1 + 48 T + 13297 T^{2} + 123072 T^{3} + 79863648 T^{4} + 123072 p^{2} T^{5} + 13297 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 83 | \( 1 - 682176 T^{3} - 7433372 T^{4} + 4619013696 T^{5} + 232682047488 T^{6} + 1985641063296 T^{7} - 3794632805852922 T^{8} + 1985641063296 p^{2} T^{9} + 232682047488 p^{4} T^{10} + 4619013696 p^{6} T^{11} - 7433372 p^{8} T^{12} - 682176 p^{10} T^{13} + p^{16} T^{16} \) | |
| 89 | \( 1 + 228 T + 25992 T^{2} + 1425324 T^{3} - 61488752 T^{4} - 10229949324 T^{5} + 281561448600 T^{6} + 199104969172284 T^{7} + 26752203506971038 T^{8} + 199104969172284 p^{2} T^{9} + 281561448600 p^{4} T^{10} - 10229949324 p^{6} T^{11} - 61488752 p^{8} T^{12} + 1425324 p^{10} T^{13} + 25992 p^{12} T^{14} + 228 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 + 2 T + 2 T^{2} + 483508 T^{3} - 63264331 T^{4} - 2869633772 T^{5} + 111277254150 T^{6} - 2613743219514 T^{7} - 3483223010579036 T^{8} - 2613743219514 p^{2} T^{9} + 111277254150 p^{4} T^{10} - 2869633772 p^{6} T^{11} - 63264331 p^{8} T^{12} + 483508 p^{10} T^{13} + 2 p^{12} T^{14} + 2 p^{14} T^{15} + p^{16} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.02318560853954895734545260120, −3.94078752284991713260052581182, −3.90889200725411999851875399776, −3.74132396322603959083871789609, −3.70196677002572146924464842222, −3.36901847567596999634364739654, −3.24720837056617210528480966251, −3.16877476476295210808783124283, −3.12654457106438234719432868712, −2.89916225596286213654875196626, −2.72900146444744760621923003056, −2.69951357028327427338042321682, −2.69041427911752305083973963186, −2.28229043088390759545423249054, −2.08418164388050280273921943061, −1.84524228280386929208126536098, −1.84436261756017953325599796947, −1.73413321612213867743297475562, −1.66473470224766811959789496705, −1.29939838527822643140746740837, −1.15895998300642378135759331621, −0.957410138347157691584884000263, −0.69105480329705321230753321192, −0.55946978424572345367654206928, −0.28999223834870131796363760876, 0.28999223834870131796363760876, 0.55946978424572345367654206928, 0.69105480329705321230753321192, 0.957410138347157691584884000263, 1.15895998300642378135759331621, 1.29939838527822643140746740837, 1.66473470224766811959789496705, 1.73413321612213867743297475562, 1.84436261756017953325599796947, 1.84524228280386929208126536098, 2.08418164388050280273921943061, 2.28229043088390759545423249054, 2.69041427911752305083973963186, 2.69951357028327427338042321682, 2.72900146444744760621923003056, 2.89916225596286213654875196626, 3.12654457106438234719432868712, 3.16877476476295210808783124283, 3.24720837056617210528480966251, 3.36901847567596999634364739654, 3.70196677002572146924464842222, 3.74132396322603959083871789609, 3.90889200725411999851875399776, 3.94078752284991713260052581182, 4.02318560853954895734545260120
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.