Dirichlet series
| L(s) = 1 | − 3·2-s − 2·3-s − 4-s + 6·6-s + 8·7-s + 10·8-s − 10·9-s − 5·11-s + 2·12-s − 9·13-s − 24·14-s − 3·16-s + 17-s + 30·18-s − 4·19-s − 16·21-s + 15·22-s + 8·23-s − 20·24-s + 27·26-s + 25·27-s − 8·28-s − 5·29-s − 31-s − 8·32-s + 10·33-s − 3·34-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s − 1/2·4-s + 2.44·6-s + 3.02·7-s + 3.53·8-s − 3.33·9-s − 1.50·11-s + 0.577·12-s − 2.49·13-s − 6.41·14-s − 3/4·16-s + 0.242·17-s + 7.07·18-s − 0.917·19-s − 3.49·21-s + 3.19·22-s + 1.66·23-s − 4.08·24-s + 5.29·26-s + 4.81·27-s − 1.51·28-s − 0.928·29-s − 0.179·31-s − 1.41·32-s + 1.74·33-s − 0.514·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{16} \cdot 7^{8} \cdot 23^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.13852\times 10^{12}\) |
| Root analytic conductor: | \(5.66919\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 5^{16} \cdot 7^{8} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 7 | \( ( 1 - T )^{8} \) | |
| 23 | \( ( 1 - T )^{8} \) | |
| good | 2 | \( 1 + 3 T + 5 p T^{2} + 23 T^{3} + 13 p^{2} T^{4} + 3 p^{5} T^{5} + 43 p^{2} T^{6} + 267 T^{7} + 407 T^{8} + 267 p T^{9} + 43 p^{4} T^{10} + 3 p^{8} T^{11} + 13 p^{6} T^{12} + 23 p^{5} T^{13} + 5 p^{7} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 + 2 T + 14 T^{2} + 23 T^{3} + 103 T^{4} + 152 T^{5} + 511 T^{6} + 650 T^{7} + 1787 T^{8} + 650 p T^{9} + 511 p^{2} T^{10} + 152 p^{3} T^{11} + 103 p^{4} T^{12} + 23 p^{5} T^{13} + 14 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 5 T + 70 T^{2} + 288 T^{3} + 2261 T^{4} + 7814 T^{5} + 44648 T^{6} + 129823 T^{7} + 592081 T^{8} + 129823 p T^{9} + 44648 p^{2} T^{10} + 7814 p^{3} T^{11} + 2261 p^{4} T^{12} + 288 p^{5} T^{13} + 70 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 9 T + 90 T^{2} + 552 T^{3} + 3242 T^{4} + 15387 T^{5} + 68244 T^{6} + 272378 T^{7} + 1016775 T^{8} + 272378 p T^{9} + 68244 p^{2} T^{10} + 15387 p^{3} T^{11} + 3242 p^{4} T^{12} + 552 p^{5} T^{13} + 90 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - T + 103 T^{2} - 159 T^{3} + 4808 T^{4} - 9344 T^{5} + 137670 T^{6} - 281203 T^{7} + 2746785 T^{8} - 281203 p T^{9} + 137670 p^{2} T^{10} - 9344 p^{3} T^{11} + 4808 p^{4} T^{12} - 159 p^{5} T^{13} + 103 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 4 T + 91 T^{2} + 340 T^{3} + 3601 T^{4} + 13255 T^{5} + 87650 T^{6} + 332944 T^{7} + 1718231 T^{8} + 332944 p T^{9} + 87650 p^{2} T^{10} + 13255 p^{3} T^{11} + 3601 p^{4} T^{12} + 340 p^{5} T^{13} + 91 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 5 T + 93 T^{2} + 333 T^{3} + 5333 T^{4} + 16991 T^{5} + 225916 T^{6} + 634890 T^{7} + 7414465 T^{8} + 634890 p T^{9} + 225916 p^{2} T^{10} + 16991 p^{3} T^{11} + 5333 p^{4} T^{12} + 333 p^{5} T^{13} + 93 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + T + 100 T^{2} - 148 T^{3} + 4385 T^{4} - 19004 T^{5} + 119190 T^{6} - 1048099 T^{7} + 2877455 T^{8} - 1048099 p T^{9} + 119190 p^{2} T^{10} - 19004 p^{3} T^{11} + 4385 p^{4} T^{12} - 148 p^{5} T^{13} + 100 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 18 T + 376 T^{2} + 4505 T^{3} + 54939 T^{4} + 489840 T^{5} + 4312804 T^{6} + 29910143 T^{7} + 203070623 T^{8} + 29910143 p T^{9} + 4312804 p^{2} T^{10} + 489840 p^{3} T^{11} + 54939 p^{4} T^{12} + 4505 p^{5} T^{13} + 376 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + T + 218 T^{2} + 322 T^{3} + 23375 T^{4} + 37522 T^{5} + 1614950 T^{6} + 2440663 T^{7} + 78223615 T^{8} + 2440663 p T^{9} + 1614950 p^{2} T^{10} + 37522 p^{3} T^{11} + 23375 p^{4} T^{12} + 322 p^{5} T^{13} + 218 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 20 T + 370 T^{2} + 4633 T^{3} + 53740 T^{4} + 509876 T^{5} + 4469206 T^{6} + 33824174 T^{7} + 236968703 T^{8} + 33824174 p T^{9} + 4469206 p^{2} T^{10} + 509876 p^{3} T^{11} + 53740 p^{4} T^{12} + 4633 p^{5} T^{13} + 370 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 10 T + 203 T^{2} + 2104 T^{3} + 23744 T^{4} + 210815 T^{5} + 1848114 T^{6} + 13961441 T^{7} + 101566631 T^{8} + 13961441 p T^{9} + 1848114 p^{2} T^{10} + 210815 p^{3} T^{11} + 23744 p^{4} T^{12} + 2104 p^{5} T^{13} + 203 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 11 T + 185 T^{2} + 1808 T^{3} + 21781 T^{4} + 178945 T^{5} + 1785701 T^{6} + 12903950 T^{7} + 107340417 T^{8} + 12903950 p T^{9} + 1785701 p^{2} T^{10} + 178945 p^{3} T^{11} + 21781 p^{4} T^{12} + 1808 p^{5} T^{13} + 185 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 20 T + 491 T^{2} - 6579 T^{3} + 96053 T^{4} - 991671 T^{5} + 10783675 T^{6} - 90024096 T^{7} + 781129041 T^{8} - 90024096 p T^{9} + 10783675 p^{2} T^{10} - 991671 p^{3} T^{11} + 96053 p^{4} T^{12} - 6579 p^{5} T^{13} + 491 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 6 T + 156 T^{2} + 1025 T^{3} + 19201 T^{4} + 111096 T^{5} + 1701891 T^{6} + 8894564 T^{7} + 113263297 T^{8} + 8894564 p T^{9} + 1701891 p^{2} T^{10} + 111096 p^{3} T^{11} + 19201 p^{4} T^{12} + 1025 p^{5} T^{13} + 156 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 23 T + 411 T^{2} + 5199 T^{3} + 61873 T^{4} + 653083 T^{5} + 6639648 T^{6} + 60135256 T^{7} + 519181851 T^{8} + 60135256 p T^{9} + 6639648 p^{2} T^{10} + 653083 p^{3} T^{11} + 61873 p^{4} T^{12} + 5199 p^{5} T^{13} + 411 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 3 T + 313 T^{2} - 317 T^{3} + 691 p T^{4} - 8243 T^{5} + 5430652 T^{6} - 422566 T^{7} + 449286567 T^{8} - 422566 p T^{9} + 5430652 p^{2} T^{10} - 8243 p^{3} T^{11} + 691 p^{5} T^{12} - 317 p^{5} T^{13} + 313 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 8 T + 242 T^{2} - 1479 T^{3} + 32419 T^{4} - 179732 T^{5} + 3196385 T^{6} - 16886574 T^{7} + 259809365 T^{8} - 16886574 p T^{9} + 3196385 p^{2} T^{10} - 179732 p^{3} T^{11} + 32419 p^{4} T^{12} - 1479 p^{5} T^{13} + 242 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 4 T + 560 T^{2} - 2142 T^{3} + 141251 T^{4} - 497005 T^{5} + 21068076 T^{6} - 64753492 T^{7} + 2039964787 T^{8} - 64753492 p T^{9} + 21068076 p^{2} T^{10} - 497005 p^{3} T^{11} + 141251 p^{4} T^{12} - 2142 p^{5} T^{13} + 560 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 4 T + 271 T^{2} + 1905 T^{3} + 42756 T^{4} + 390631 T^{5} + 4863288 T^{6} + 48279102 T^{7} + 452154785 T^{8} + 48279102 p T^{9} + 4863288 p^{2} T^{10} + 390631 p^{3} T^{11} + 42756 p^{4} T^{12} + 1905 p^{5} T^{13} + 271 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 17 T + 720 T^{2} + 9700 T^{3} + 221878 T^{4} + 2426901 T^{5} + 38946962 T^{6} + 347718674 T^{7} + 4307242151 T^{8} + 347718674 p T^{9} + 38946962 p^{2} T^{10} + 2426901 p^{3} T^{11} + 221878 p^{4} T^{12} + 9700 p^{5} T^{13} + 720 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 41 T + 1234 T^{2} + 26542 T^{3} + 486246 T^{4} + 7410777 T^{5} + 100057472 T^{6} + 1171754020 T^{7} + 12324003755 T^{8} + 1171754020 p T^{9} + 100057472 p^{2} T^{10} + 7410777 p^{3} T^{11} + 486246 p^{4} T^{12} + 26542 p^{5} T^{13} + 1234 p^{6} T^{14} + 41 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.05619372816312579362408381056, −3.75059285259137553677090641206, −3.73049012491806388691321692351, −3.43059546510919806947925940251, −3.40947883128762172890702470771, −3.37918624904589597934577407833, −3.27155023795854015866686045172, −2.99152365125885982708330251819, −2.88482499121934836500873384066, −2.77337492821178770879108390772, −2.73474623944536536469891814455, −2.71536178896703843840147592107, −2.50444561193970309639747310080, −2.38085643370524670792032556126, −2.31520671904292126459761713069, −2.20381889153317641860109514603, −1.97760795215257796372923160659, −1.93791232230168940390655716939, −1.56326256728185583162091013532, −1.48196137297642158962142068057, −1.37609196464180672398154243850, −1.22316272588483072899083759951, −1.08383078891166677129405575408, −1.07837033717018982955896379677, −0.927013848379426436195489717566, 0, 0, 0, 0, 0, 0, 0, 0, 0.927013848379426436195489717566, 1.07837033717018982955896379677, 1.08383078891166677129405575408, 1.22316272588483072899083759951, 1.37609196464180672398154243850, 1.48196137297642158962142068057, 1.56326256728185583162091013532, 1.93791232230168940390655716939, 1.97760795215257796372923160659, 2.20381889153317641860109514603, 2.31520671904292126459761713069, 2.38085643370524670792032556126, 2.50444561193970309639747310080, 2.71536178896703843840147592107, 2.73474623944536536469891814455, 2.77337492821178770879108390772, 2.88482499121934836500873384066, 2.99152365125885982708330251819, 3.27155023795854015866686045172, 3.37918624904589597934577407833, 3.40947883128762172890702470771, 3.43059546510919806947925940251, 3.73049012491806388691321692351, 3.75059285259137553677090641206, 4.05619372816312579362408381056
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.