Dirichlet series
| L(s) = 1 | + 2-s + 2·3-s + 3·4-s + 2·6-s − 4·7-s + 2·8-s + 9-s + 16·11-s + 6·12-s + 8·13-s − 4·14-s + 4·16-s + 17-s + 18-s − 5·19-s − 8·21-s + 16·22-s − 7·23-s + 4·24-s + 8·26-s − 12·28-s + 5·29-s − 19·31-s + 9·32-s + 32·33-s + 34-s + 3·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 3/2·4-s + 0.816·6-s − 1.51·7-s + 0.707·8-s + 1/3·9-s + 4.82·11-s + 1.73·12-s + 2.21·13-s − 1.06·14-s + 16-s + 0.242·17-s + 0.235·18-s − 1.14·19-s − 1.74·21-s + 3.41·22-s − 1.45·23-s + 0.816·24-s + 1.56·26-s − 2.26·28-s + 0.928·29-s − 3.41·31-s + 1.59·32-s + 5.57·33-s + 0.171·34-s + 1/2·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6463.48\) |
| Root analytic conductor: | \(1.73043\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{24} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7.654048971\) |
| \(L(\frac12)\) | \(\approx\) | \(7.654048971\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 5 | \( 1 \) | |
| good | 2 | \( ( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} ) \) |
| 7 | \( ( 1 + 2 T + 12 T^{2} + 15 T^{3} + 61 T^{4} + 15 p T^{5} + 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 11 | \( 1 - 16 T + 105 T^{2} - 340 T^{3} + 390 T^{4} + 1332 T^{5} - 9707 T^{6} + 42730 T^{7} - 156045 T^{8} + 42730 p T^{9} - 9707 p^{2} T^{10} + 1332 p^{3} T^{11} + 390 p^{4} T^{12} - 340 p^{5} T^{13} + 105 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 8 T + 17 T^{2} - 14 T^{3} + 276 T^{4} - 578 T^{5} - 4815 T^{6} + 18048 T^{7} - 25061 T^{8} + 18048 p T^{9} - 4815 p^{2} T^{10} - 578 p^{3} T^{11} + 276 p^{4} T^{12} - 14 p^{5} T^{13} + 17 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - T - 27 T^{2} + 38 T^{3} + 591 T^{4} - 456 T^{5} - 11720 T^{6} + 3109 T^{7} + 167979 T^{8} + 3109 p T^{9} - 11720 p^{2} T^{10} - 456 p^{3} T^{11} + 591 p^{4} T^{12} + 38 p^{5} T^{13} - 27 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 5 T - 33 T^{2} - 265 T^{3} - 47 T^{4} + 6980 T^{5} + 26814 T^{6} - 68150 T^{7} - 765695 T^{8} - 68150 p T^{9} + 26814 p^{2} T^{10} + 6980 p^{3} T^{11} - 47 p^{4} T^{12} - 265 p^{5} T^{13} - 33 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 7 T + 12 T^{2} + 31 T^{3} + 1026 T^{4} + 672 T^{5} - 34550 T^{6} - 127672 T^{7} - 3411 T^{8} - 127672 p T^{9} - 34550 p^{2} T^{10} + 672 p^{3} T^{11} + 1026 p^{4} T^{12} + 31 p^{5} T^{13} + 12 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 5 T + 12 T^{2} + 110 T^{3} - 322 T^{4} + 4795 T^{5} - 5456 T^{6} + 22500 T^{7} + 487955 T^{8} + 22500 p T^{9} - 5456 p^{2} T^{10} + 4795 p^{3} T^{11} - 322 p^{4} T^{12} + 110 p^{5} T^{13} + 12 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 19 T + 110 T^{2} - 455 T^{3} - 10080 T^{4} - 55148 T^{5} - 24492 T^{6} + 1584870 T^{7} + 12593785 T^{8} + 1584870 p T^{9} - 24492 p^{2} T^{10} - 55148 p^{3} T^{11} - 10080 p^{4} T^{12} - 455 p^{5} T^{13} + 110 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - T + 3 T^{2} - 42 T^{3} + 2811 T^{4} - 3486 T^{5} - 59280 T^{6} - 126501 T^{7} + 3465939 T^{8} - 126501 p T^{9} - 59280 p^{2} T^{10} - 3486 p^{3} T^{11} + 2811 p^{4} T^{12} - 42 p^{5} T^{13} + 3 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( ( 1 + 7 T + 28 T^{2} + 389 T^{3} + 3975 T^{4} + 389 p T^{5} + 28 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 43 | \( ( 1 + 16 T + 233 T^{2} + 2040 T^{3} + 16241 T^{4} + 2040 p T^{5} + 233 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - T - 62 T^{2} + 78 T^{3} + 3466 T^{4} - 4691 T^{5} - 155900 T^{6} + 95604 T^{7} + 3265179 T^{8} + 95604 p T^{9} - 155900 p^{2} T^{10} - 4691 p^{3} T^{11} + 3466 p^{4} T^{12} + 78 p^{5} T^{13} - 62 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 3 T - 63 T^{2} + 6 T^{3} + 4741 T^{4} + 3522 T^{5} - 240240 T^{6} + 209553 T^{7} + 7555219 T^{8} + 209553 p T^{9} - 240240 p^{2} T^{10} + 3522 p^{3} T^{11} + 4741 p^{4} T^{12} + 6 p^{5} T^{13} - 63 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 30 T + 377 T^{2} - 2475 T^{3} + 8643 T^{4} - 1155 T^{5} - 582521 T^{6} + 10821600 T^{7} - 108687625 T^{8} + 10821600 p T^{9} - 582521 p^{2} T^{10} - 1155 p^{3} T^{11} + 8643 p^{4} T^{12} - 2475 p^{5} T^{13} + 377 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 14 T + 35 T^{2} + 110 T^{3} + 8070 T^{4} + 33812 T^{5} - 419217 T^{6} - 1871430 T^{7} + 12043555 T^{8} - 1871430 p T^{9} - 419217 p^{2} T^{10} + 33812 p^{3} T^{11} + 8070 p^{4} T^{12} + 110 p^{5} T^{13} + 35 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 4 T - 97 T^{2} - 122 T^{3} + 9681 T^{4} + 8074 T^{5} - 723795 T^{6} - 600576 T^{7} + 39163204 T^{8} - 600576 p T^{9} - 723795 p^{2} T^{10} + 8074 p^{3} T^{11} + 9681 p^{4} T^{12} - 122 p^{5} T^{13} - 97 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 21 T + 305 T^{2} - 2480 T^{3} + 13165 T^{4} + 10682 T^{5} - 810942 T^{6} + 9320005 T^{7} - 80224815 T^{8} + 9320005 p T^{9} - 810942 p^{2} T^{10} + 10682 p^{3} T^{11} + 13165 p^{4} T^{12} - 2480 p^{5} T^{13} + 305 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 2 T - 163 T^{2} + 986 T^{3} + 8481 T^{4} - 176518 T^{5} + 938895 T^{6} + 6925578 T^{7} - 156207836 T^{8} + 6925578 p T^{9} + 938895 p^{2} T^{10} - 176518 p^{3} T^{11} + 8481 p^{4} T^{12} + 986 p^{5} T^{13} - 163 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 30 T + 447 T^{2} + 5690 T^{3} + 63618 T^{4} + 615980 T^{5} + 6355819 T^{6} + 63162750 T^{7} + 565067755 T^{8} + 63162750 p T^{9} + 6355819 p^{2} T^{10} + 615980 p^{3} T^{11} + 63618 p^{4} T^{12} + 5690 p^{5} T^{13} + 447 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 2 T - 123 T^{2} + 626 T^{3} + 1416 T^{4} - 170328 T^{5} + 1146775 T^{6} + 9202888 T^{7} - 131919081 T^{8} + 9202888 p T^{9} + 1146775 p^{2} T^{10} - 170328 p^{3} T^{11} + 1416 p^{4} T^{12} + 626 p^{5} T^{13} - 123 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 77 T^{2} + 700 T^{3} + 12658 T^{4} + 121100 T^{5} + 1037049 T^{6} + 10883950 T^{7} + 150329955 T^{8} + 10883950 p T^{9} + 1037049 p^{2} T^{10} + 121100 p^{3} T^{11} + 12658 p^{4} T^{12} + 700 p^{5} T^{13} + 77 p^{6} T^{14} + p^{8} T^{16} \) | |
| 97 | \( 1 - 6 T - 187 T^{2} + 1878 T^{3} + 3816 T^{4} - 286836 T^{5} + 2735335 T^{6} + 15219804 T^{7} - 421030321 T^{8} + 15219804 p T^{9} + 2735335 p^{2} T^{10} - 286836 p^{3} T^{11} + 3816 p^{4} T^{12} + 1878 p^{5} T^{13} - 187 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.04196657838640854755302716699, −4.84323495675934446305743453916, −4.69869912606995568510904540273, −4.69213924341779913792347105502, −4.15309340866185465365070616535, −4.14676490047641558036677464531, −4.05809881548241987104843882082, −3.86668016565621147098537968395, −3.85187819064306511500516374268, −3.72539706611156850055229216968, −3.59593659762273429428076372639, −3.25944412760552519783532409088, −3.25481643438509016346216215979, −3.21770190189098572982615842922, −3.12317580800354826047444026278, −2.78029299600114639528953453920, −2.50045347786238596003330099131, −2.27561653009304127252283521021, −2.04063584341240218891677689562, −1.70254512412207936223398275699, −1.70222580872542581281915417029, −1.55124967005318200308572702618, −1.29728451353331249801466750643, −1.24229444944271261465591660443, −0.35125395380730247462167170189, 0.35125395380730247462167170189, 1.24229444944271261465591660443, 1.29728451353331249801466750643, 1.55124967005318200308572702618, 1.70222580872542581281915417029, 1.70254512412207936223398275699, 2.04063584341240218891677689562, 2.27561653009304127252283521021, 2.50045347786238596003330099131, 2.78029299600114639528953453920, 3.12317580800354826047444026278, 3.21770190189098572982615842922, 3.25481643438509016346216215979, 3.25944412760552519783532409088, 3.59593659762273429428076372639, 3.72539706611156850055229216968, 3.85187819064306511500516374268, 3.86668016565621147098537968395, 4.05809881548241987104843882082, 4.14676490047641558036677464531, 4.15309340866185465365070616535, 4.69213924341779913792347105502, 4.69869912606995568510904540273, 4.84323495675934446305743453916, 5.04196657838640854755302716699
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.