Dirichlet series
| L(s) = 1 | + 4·2-s − 3-s + 6·4-s − 3·5-s − 4·6-s − 7-s + 3·9-s − 12·10-s + 6·11-s − 6·12-s − 2·13-s − 4·14-s + 3·15-s − 15·16-s − 15·17-s + 12·18-s + 19-s − 18·20-s + 21-s + 24·22-s − 4·23-s + 17·25-s − 8·26-s − 8·27-s − 6·28-s + 12·29-s + 12·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 0.577·3-s + 3·4-s − 1.34·5-s − 1.63·6-s − 0.377·7-s + 9-s − 3.79·10-s + 1.80·11-s − 1.73·12-s − 0.554·13-s − 1.06·14-s + 0.774·15-s − 3.75·16-s − 3.63·17-s + 2.82·18-s + 0.229·19-s − 4.02·20-s + 0.218·21-s + 5.11·22-s − 0.834·23-s + 17/5·25-s − 1.56·26-s − 1.53·27-s − 1.13·28-s + 2.22·29-s + 2.19·30-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
| \(L(1)\) | \(\approx\) | \(0.7275286447\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7275286447\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + T - 2 T^{2} - 17 T^{3} - 17 T^{4} - 17 p T^{5} - 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | |
| 23 | \( ( 1 + T + T^{2} )^{4} \) | |
| good | 3 | \( 1 + T - 2 T^{2} + p T^{3} - 2 p^{2} T^{5} + p^{3} T^{6} + 2 p^{3} T^{7} - p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{13} - 2 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 3 T - 8 T^{2} - 29 T^{3} + 46 T^{4} + 142 T^{5} - 241 T^{6} - 46 p T^{7} + 1529 T^{8} - 46 p^{2} T^{9} - 241 p^{2} T^{10} + 142 p^{3} T^{11} + 46 p^{4} T^{12} - 29 p^{5} T^{13} - 8 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 6 T - 8 T^{2} + 38 T^{3} + 367 T^{4} - 349 T^{5} - 5689 T^{6} + 6125 T^{7} + 35693 T^{8} + 6125 p T^{9} - 5689 p^{2} T^{10} - 349 p^{3} T^{11} + 367 p^{4} T^{12} + 38 p^{5} T^{13} - 8 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( ( 1 + T + 28 T^{2} - 31 T^{3} + 341 T^{4} - 31 p T^{5} + 28 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 17 | \( 1 + 15 T + 79 T^{2} + 328 T^{3} + 2842 T^{4} + 16483 T^{5} + 55430 T^{6} + 283000 T^{7} + 1563149 T^{8} + 283000 p T^{9} + 55430 p^{2} T^{10} + 16483 p^{3} T^{11} + 2842 p^{4} T^{12} + 328 p^{5} T^{13} + 79 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - T - 30 T^{2} + 33 T^{3} + 414 T^{4} - 30 p T^{5} + 6833 T^{6} + 3166 T^{7} - 217755 T^{8} + 3166 p T^{9} + 6833 p^{2} T^{10} - 30 p^{4} T^{11} + 414 p^{4} T^{12} + 33 p^{5} T^{13} - 30 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( ( 1 - 6 T + 101 T^{2} - 422 T^{3} + 4095 T^{4} - 422 p T^{5} + 101 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 31 | \( 1 + 8 T - 54 T^{2} - 358 T^{3} + 3409 T^{4} + 12087 T^{5} - 137077 T^{6} - 5237 p T^{7} + 4535181 T^{8} - 5237 p^{2} T^{9} - 137077 p^{2} T^{10} + 12087 p^{3} T^{11} + 3409 p^{4} T^{12} - 358 p^{5} T^{13} - 54 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 8 T + 36 T^{2} - 350 T^{3} - 4645 T^{4} - 31473 T^{5} - 13643 T^{6} + 1016387 T^{7} + 10304163 T^{8} + 1016387 p T^{9} - 13643 p^{2} T^{10} - 31473 p^{3} T^{11} - 4645 p^{4} T^{12} - 350 p^{5} T^{13} + 36 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( ( 1 - 9 T + 146 T^{2} - 801 T^{3} + 8043 T^{4} - 801 p T^{5} + 146 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 43 | \( ( 1 - 14 T + 154 T^{2} - 1435 T^{3} + 10313 T^{4} - 1435 p T^{5} + 154 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 9 T - 62 T^{2} - 645 T^{3} + 2092 T^{4} + 9606 T^{5} - 225497 T^{6} + 126876 T^{7} + 16283041 T^{8} + 126876 p T^{9} - 225497 p^{2} T^{10} + 9606 p^{3} T^{11} + 2092 p^{4} T^{12} - 645 p^{5} T^{13} - 62 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 3 T - 38 T^{2} + 7 p T^{3} - 1292 T^{4} - 9202 T^{5} + 160427 T^{6} - 266956 T^{7} - 7419895 T^{8} - 266956 p T^{9} + 160427 p^{2} T^{10} - 9202 p^{3} T^{11} - 1292 p^{4} T^{12} + 7 p^{6} T^{13} - 38 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 12 T - 92 T^{2} - 1206 T^{3} + 10663 T^{4} + 96285 T^{5} - 620039 T^{6} - 1897077 T^{7} + 45496741 T^{8} - 1897077 p T^{9} - 620039 p^{2} T^{10} + 96285 p^{3} T^{11} + 10663 p^{4} T^{12} - 1206 p^{5} T^{13} - 92 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 11 T + 75 T^{2} + 1812 T^{3} + 18780 T^{4} + 138711 T^{5} + 1643030 T^{6} + 14235346 T^{7} + 97088337 T^{8} + 14235346 p T^{9} + 1643030 p^{2} T^{10} + 138711 p^{3} T^{11} + 18780 p^{4} T^{12} + 1812 p^{5} T^{13} + 75 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - T - 183 T^{2} + 370 T^{3} + 17012 T^{4} - 36393 T^{5} - 1338206 T^{6} + 1113914 T^{7} + 97795467 T^{8} + 1113914 p T^{9} - 1338206 p^{2} T^{10} - 36393 p^{3} T^{11} + 17012 p^{4} T^{12} + 370 p^{5} T^{13} - 183 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 + 3 T + 158 T^{2} + 1059 T^{3} + 11985 T^{4} + 1059 p T^{5} + 158 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( 1 - 4 T - 186 T^{2} + 818 T^{3} + 17251 T^{4} - 61167 T^{5} - 20881 p T^{6} + 1610383 T^{7} + 130485651 T^{8} + 1610383 p T^{9} - 20881 p^{3} T^{10} - 61167 p^{3} T^{11} + 17251 p^{4} T^{12} + 818 p^{5} T^{13} - 186 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 5 T + 93 T^{2} - 526 T^{3} - 8888 T^{4} - 129447 T^{5} + 10334 T^{6} + 2555818 T^{7} + 126384423 T^{8} + 2555818 p T^{9} + 10334 p^{2} T^{10} - 129447 p^{3} T^{11} - 8888 p^{4} T^{12} - 526 p^{5} T^{13} + 93 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 - 12 T + 311 T^{2} - 2498 T^{3} + 36819 T^{4} - 2498 p T^{5} + 311 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 27 T + 202 T^{2} - 423 T^{3} + 2692 T^{4} + 342126 T^{5} + 3711787 T^{6} + 10342926 T^{7} - 46198187 T^{8} + 10342926 p T^{9} + 3711787 p^{2} T^{10} + 342126 p^{3} T^{11} + 2692 p^{4} T^{12} - 423 p^{5} T^{13} + 202 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( ( 1 - 2 T + 169 T^{2} + 18 T^{3} + 16395 T^{4} + 18 p T^{5} + 169 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
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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
−5.02879610486779812112847419202, −4.92300999803755893408245538827, −4.89130851864802570996357816946, −4.73578502500176130592504879099, −4.49644037192754578681410660952, −4.35485382163030707683636724915, −4.24250361468833011411746974426, −4.24249903856706000876070794589, −4.10063005290357042320431578008, −4.08349466336572771394109098486, −3.84382252835713366516145792981, −3.56278619369183984374335116871, −3.51459032865234843216523432358, −3.17996774258039127593581343547, −3.11638550737014485255222719456, −3.06088359487307738566057509435, −2.58045569337166277537975048403, −2.51725599359964399866706738260, −2.26223143565447754276384065967, −2.15771577716793156206957020363, −2.08469780738566910983275004200, −1.28319702701536308953631817514, −1.16188430223833592435471202130, −1.00106522908241647565033960675, −0.14032784609561434340740717116, 0.14032784609561434340740717116, 1.00106522908241647565033960675, 1.16188430223833592435471202130, 1.28319702701536308953631817514, 2.08469780738566910983275004200, 2.15771577716793156206957020363, 2.26223143565447754276384065967, 2.51725599359964399866706738260, 2.58045569337166277537975048403, 3.06088359487307738566057509435, 3.11638550737014485255222719456, 3.17996774258039127593581343547, 3.51459032865234843216523432358, 3.56278619369183984374335116871, 3.84382252835713366516145792981, 4.08349466336572771394109098486, 4.10063005290357042320431578008, 4.24249903856706000876070794589, 4.24250361468833011411746974426, 4.35485382163030707683636724915, 4.49644037192754578681410660952, 4.73578502500176130592504879099, 4.89130851864802570996357816946, 4.92300999803755893408245538827, 5.02879610486779812112847419202
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.