Dirichlet series
| L(s) = 1 | + 4·2-s − 3-s + 6·4-s + 5·5-s − 4·6-s − 3·7-s + 7·9-s + 20·10-s + 2·11-s − 6·12-s + 14·13-s − 12·14-s − 5·15-s − 15·16-s + 7·17-s + 28·18-s + 19-s + 30·20-s + 3·21-s + 8·22-s + 4·23-s + 13·25-s + 56·26-s − 4·27-s − 18·28-s − 12·29-s − 20·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 0.577·3-s + 3·4-s + 2.23·5-s − 1.63·6-s − 1.13·7-s + 7/3·9-s + 6.32·10-s + 0.603·11-s − 1.73·12-s + 3.88·13-s − 3.20·14-s − 1.29·15-s − 3.75·16-s + 1.69·17-s + 6.59·18-s + 0.229·19-s + 6.70·20-s + 0.654·21-s + 1.70·22-s + 0.834·23-s + 13/5·25-s + 10.9·26-s − 0.769·27-s − 3.40·28-s − 2.22·29-s − 3.65·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 7^{8} \cdot 23^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1910.13\) |
| Root analytic conductor: | \(1.60349\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 7^{8} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(25.44225361\) |
| \(L(\frac12)\) | \(\approx\) | \(25.44225361\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{4} \) |
| 7 | \( 1 + 3 T - 4 T^{2} - 3 T^{3} + 57 T^{4} - 3 p T^{5} - 4 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| 23 | \( ( 1 - T + T^{2} )^{4} \) | |
| good | 3 | \( 1 + T - 2 p T^{2} - p^{2} T^{3} + 16 T^{4} + 10 p T^{5} - p^{2} T^{6} - 46 T^{7} - 35 T^{8} - 46 p T^{9} - p^{4} T^{10} + 10 p^{4} T^{11} + 16 p^{4} T^{12} - p^{7} T^{13} - 2 p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - p T + 12 T^{2} - 17 T^{3} - 4 T^{4} + 98 T^{5} - 63 T^{6} - 402 T^{7} + 1329 T^{8} - 402 p T^{9} - 63 p^{2} T^{10} + 98 p^{3} T^{11} - 4 p^{4} T^{12} - 17 p^{5} T^{13} + 12 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 2 T - 4 T^{2} - 10 T^{3} + 97 T^{4} - 213 T^{5} + 2191 T^{6} - 3953 T^{7} - 10315 T^{8} - 3953 p T^{9} + 2191 p^{2} T^{10} - 213 p^{3} T^{11} + 97 p^{4} T^{12} - 10 p^{5} T^{13} - 4 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( ( 1 - 7 T + 4 p T^{2} - 235 T^{3} + 1005 T^{4} - 235 p T^{5} + 4 p^{3} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 17 | \( 1 - 7 T - 21 T^{2} + 188 T^{3} + 716 T^{4} - 4139 T^{5} - 690 p T^{6} + 16890 T^{7} + 307917 T^{8} + 16890 p T^{9} - 690 p^{3} T^{10} - 4139 p^{3} T^{11} + 716 p^{4} T^{12} + 188 p^{5} T^{13} - 21 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - T - 58 T^{2} + 93 T^{3} + 1888 T^{4} - 2894 T^{5} - 42573 T^{6} + 1418 p T^{7} + 818301 T^{8} + 1418 p^{2} T^{9} - 42573 p^{2} T^{10} - 2894 p^{3} T^{11} + 1888 p^{4} T^{12} + 93 p^{5} T^{13} - 58 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( ( 1 + 6 T + 77 T^{2} + 326 T^{3} + 2931 T^{4} + 326 p T^{5} + 77 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 31 | \( 1 - 4 T - 82 T^{2} + 50 T^{3} + 4757 T^{4} + 3195 T^{5} - 178581 T^{6} - 28903 T^{7} + 4991645 T^{8} - 28903 p T^{9} - 178581 p^{2} T^{10} + 3195 p^{3} T^{11} + 4757 p^{4} T^{12} + 50 p^{5} T^{13} - 82 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 78 T^{2} - 78 T^{3} + 2501 T^{4} + 4485 T^{5} - 64389 T^{6} - 85371 T^{7} + 2578455 T^{8} - 85371 p T^{9} - 64389 p^{2} T^{10} + 4485 p^{3} T^{11} + 2501 p^{4} T^{12} - 78 p^{5} T^{13} - 78 p^{6} T^{14} + p^{8} T^{16} \) | |
| 41 | \( ( 1 + 15 T + 86 T^{2} - 197 T^{3} - 3957 T^{4} - 197 p T^{5} + 86 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 43 | \( ( 1 + 12 T + 192 T^{2} + 1361 T^{3} + 12267 T^{4} + 1361 p T^{5} + 192 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 15 T + 18 T^{2} + 147 T^{3} + 4976 T^{4} - 8514 T^{5} - 378909 T^{6} + 1299060 T^{7} + 4716345 T^{8} + 1299060 p T^{9} - 378909 p^{2} T^{10} - 8514 p^{3} T^{11} + 4976 p^{4} T^{12} + 147 p^{5} T^{13} + 18 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 21 T + 112 T^{2} + 177 T^{3} + 9028 T^{4} + 87678 T^{5} + 46537 T^{6} + 1169796 T^{7} + 32672125 T^{8} + 1169796 p T^{9} + 46537 p^{2} T^{10} + 87678 p^{3} T^{11} + 9028 p^{4} T^{12} + 177 p^{5} T^{13} + 112 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 32 T + 436 T^{2} - 4598 T^{3} + 54583 T^{4} - 8889 p T^{5} + 3770021 T^{6} - 31471125 T^{7} + 277479917 T^{8} - 31471125 p T^{9} + 3770021 p^{2} T^{10} - 8889 p^{4} T^{11} + 54583 p^{4} T^{12} - 4598 p^{5} T^{13} + 436 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 3 T - 193 T^{2} + 512 T^{3} + 21898 T^{4} - 43643 T^{5} - 1766790 T^{6} + 1209116 T^{7} + 116185233 T^{8} + 1209116 p T^{9} - 1766790 p^{2} T^{10} - 43643 p^{3} T^{11} + 21898 p^{4} T^{12} + 512 p^{5} T^{13} - 193 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 13 T - 65 T^{2} - 2080 T^{3} - 918 T^{4} + 188955 T^{5} + 1191788 T^{6} - 5419518 T^{7} - 102818057 T^{8} - 5419518 p T^{9} + 1191788 p^{2} T^{10} + 188955 p^{3} T^{11} - 918 p^{4} T^{12} - 2080 p^{5} T^{13} - 65 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 + 7 T + 122 T^{2} + 691 T^{3} + 8049 T^{4} + 691 p T^{5} + 122 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( 1 - 16 T + 18 T^{2} + 2066 T^{3} - 15761 T^{4} - 100191 T^{5} + 1683863 T^{6} + 899791 T^{7} - 114952581 T^{8} + 899791 p T^{9} + 1683863 p^{2} T^{10} - 100191 p^{3} T^{11} - 15761 p^{4} T^{12} + 2066 p^{5} T^{13} + 18 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 3 T - 187 T^{2} - 1218 T^{3} + 15998 T^{4} + 130071 T^{5} - 815094 T^{6} - 5018052 T^{7} + 54197735 T^{8} - 5018052 p T^{9} - 815094 p^{2} T^{10} + 130071 p^{3} T^{11} + 15998 p^{4} T^{12} - 1218 p^{5} T^{13} - 187 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 - 8 T + 341 T^{2} - 1964 T^{3} + 42837 T^{4} - 1964 p T^{5} + 341 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 - 33 T + 342 T^{2} - 3055 T^{3} + 80882 T^{4} - 1054790 T^{5} + 6484681 T^{6} - 87080494 T^{7} + 1266524245 T^{8} - 87080494 p T^{9} + 6484681 p^{2} T^{10} - 1054790 p^{3} T^{11} + 80882 p^{4} T^{12} - 3055 p^{5} T^{13} + 342 p^{6} T^{14} - 33 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( ( 1 + 12 T + 145 T^{2} + 492 T^{3} + 9113 T^{4} + 492 p T^{5} + 145 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 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
−5.22769808902430597323286003764, −5.05199181143117966638187275508, −5.01857272806298741692207650838, −4.75057459156503445271222105660, −4.74418173260715118988614443431, −4.66779435698659726691586045149, −4.06397624131547766946806521497, −4.04044139002256827569200709457, −4.00665010463842970661847142593, −3.78139892080747904847609973027, −3.75604305101717251090562701037, −3.61565547089290100557666384613, −3.49893011166327208093786508804, −3.17477298290424920751192504921, −3.17276056710254795138666663564, −3.07587999470503136839862690283, −2.83570200365886714106661630867, −2.50044826295007814575563977111, −2.03026660411666755580524132901, −1.83847671995930325470646366599, −1.81499270096518832370041502758, −1.68043613993942498891911948659, −1.26669611100894962120646804937, −1.03281277489778284240708581485, −0.78785489604573850298808521694, 0.78785489604573850298808521694, 1.03281277489778284240708581485, 1.26669611100894962120646804937, 1.68043613993942498891911948659, 1.81499270096518832370041502758, 1.83847671995930325470646366599, 2.03026660411666755580524132901, 2.50044826295007814575563977111, 2.83570200365886714106661630867, 3.07587999470503136839862690283, 3.17276056710254795138666663564, 3.17477298290424920751192504921, 3.49893011166327208093786508804, 3.61565547089290100557666384613, 3.75604305101717251090562701037, 3.78139892080747904847609973027, 4.00665010463842970661847142593, 4.04044139002256827569200709457, 4.06397624131547766946806521497, 4.66779435698659726691586045149, 4.74418173260715118988614443431, 4.75057459156503445271222105660, 5.01857272806298741692207650838, 5.05199181143117966638187275508, 5.22769808902430597323286003764
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.