Dirichlet series
| L(s) = 1 | − 2-s + 4·3-s + 4·4-s − 3·5-s − 4·6-s − 2·8-s + 10·9-s + 3·10-s + 5·11-s + 16·12-s − 5·13-s − 12·15-s + 9·16-s + 11·17-s − 10·18-s + 9·19-s − 12·20-s − 5·22-s − 16·23-s − 8·24-s + 12·25-s + 5·26-s + 32·27-s − 9·29-s + 12·30-s + 11·31-s − 14·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 2.30·3-s + 2·4-s − 1.34·5-s − 1.63·6-s − 0.707·8-s + 10/3·9-s + 0.948·10-s + 1.50·11-s + 4.61·12-s − 1.38·13-s − 3.09·15-s + 9/4·16-s + 2.66·17-s − 2.35·18-s + 2.06·19-s − 2.68·20-s − 1.06·22-s − 3.33·23-s − 1.63·24-s + 12/5·25-s + 0.980·26-s + 6.15·27-s − 1.67·29-s + 2.19·30-s + 1.97·31-s − 2.47·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(7^{16} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(117740.\) |
| Root analytic conductor: | \(2.07459\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 7^{16} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(18.79352600\) |
| \(L(\frac12)\) | \(\approx\) | \(18.79352600\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - 5 T + 26 T^{2} - 75 T^{3} + 251 T^{4} - 75 p T^{5} + 26 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( 1 + T - 3 T^{2} - 5 T^{3} + 19 T^{5} + 21 T^{6} - 5 p^{2} T^{7} - 51 T^{8} - 5 p^{3} T^{9} + 21 p^{2} T^{10} + 19 p^{3} T^{11} - 5 p^{5} T^{13} - 3 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( ( 1 - 2 T + T^{2} - 2 p T^{3} + 19 T^{4} - 2 p^{2} T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 5 | \( 1 + 3 T - 3 T^{2} - 6 p T^{3} - 39 T^{4} + 6 p^{2} T^{5} + 428 T^{6} - 231 T^{7} - 2169 T^{8} - 231 p T^{9} + 428 p^{2} T^{10} + 6 p^{5} T^{11} - 39 p^{4} T^{12} - 6 p^{6} T^{13} - 3 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 5 T + 35 T^{2} + 90 T^{3} + 47 p T^{4} + 670 T^{5} + 4380 T^{6} - 8865 T^{7} + 38151 T^{8} - 8865 p T^{9} + 4380 p^{2} T^{10} + 670 p^{3} T^{11} + 47 p^{5} T^{12} + 90 p^{5} T^{13} + 35 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 11 T + 39 T^{2} - 139 T^{3} + 1249 T^{4} - 7484 T^{5} + 34098 T^{6} - 128922 T^{7} + 457683 T^{8} - 128922 p T^{9} + 34098 p^{2} T^{10} - 7484 p^{3} T^{11} + 1249 p^{4} T^{12} - 139 p^{5} T^{13} + 39 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + 1023 T^{4} - 1674 T^{5} - 12940 T^{6} + 86580 T^{7} - 253639 T^{8} + 86580 p T^{9} - 12940 p^{2} T^{10} - 1674 p^{3} T^{11} + 1023 p^{4} T^{12} - 9 p^{6} T^{13} + 43 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 8 T + 83 T^{2} + 402 T^{3} + 2555 T^{4} + 402 p T^{5} + 83 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 9 T - 22 T^{2} - 429 T^{3} - 762 T^{4} + 11754 T^{5} + 81940 T^{6} - 191700 T^{7} - 3727369 T^{8} - 191700 p T^{9} + 81940 p^{2} T^{10} + 11754 p^{3} T^{11} - 762 p^{4} T^{12} - 429 p^{5} T^{13} - 22 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 11 T + 34 T^{2} - 183 T^{3} + 3210 T^{4} - 12836 T^{5} - 31430 T^{6} - 25700 T^{7} + 2517721 T^{8} - 25700 p T^{9} - 31430 p^{2} T^{10} - 12836 p^{3} T^{11} + 3210 p^{4} T^{12} - 183 p^{5} T^{13} + 34 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 6 T - 63 T^{2} + 520 T^{3} + 1350 T^{4} - 12954 T^{5} - 20209 T^{6} + 92390 T^{7} + 1366879 T^{8} + 92390 p T^{9} - 20209 p^{2} T^{10} - 12954 p^{3} T^{11} + 1350 p^{4} T^{12} + 520 p^{5} T^{13} - 63 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 22 T + 160 T^{2} - 414 T^{3} + 2641 T^{4} - 46342 T^{5} + 334288 T^{6} - 25916 p T^{7} + 2274033 T^{8} - 25916 p^{2} T^{9} + 334288 p^{2} T^{10} - 46342 p^{3} T^{11} + 2641 p^{4} T^{12} - 414 p^{5} T^{13} + 160 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 47 | \( 1 + 7 T - 100 T^{2} - 1414 T^{3} + 286 T^{4} + 103523 T^{5} + 579636 T^{6} - 2543800 T^{7} - 42803781 T^{8} - 2543800 p T^{9} + 579636 p^{2} T^{10} + 103523 p^{3} T^{11} + 286 p^{4} T^{12} - 1414 p^{5} T^{13} - 100 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 2 T - 173 T^{2} + 964 T^{3} + 9936 T^{4} - 104742 T^{5} + 122655 T^{6} + 3229712 T^{7} - 31796061 T^{8} + 3229712 p T^{9} + 122655 p^{2} T^{10} - 104742 p^{3} T^{11} + 9936 p^{4} T^{12} + 964 p^{5} T^{13} - 173 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 25 T + 206 T^{2} - 25 T^{3} - 15220 T^{4} - 157300 T^{5} - 429276 T^{6} + 8492650 T^{7} + 113408229 T^{8} + 8492650 p T^{9} - 429276 p^{2} T^{10} - 157300 p^{3} T^{11} - 15220 p^{4} T^{12} - 25 p^{5} T^{13} + 206 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 7 T - 108 T^{2} - 1783 T^{3} + 318 T^{4} + 164342 T^{5} + 1092660 T^{6} - 5279480 T^{7} - 100824709 T^{8} - 5279480 p T^{9} + 1092660 p^{2} T^{10} + 164342 p^{3} T^{11} + 318 p^{4} T^{12} - 1783 p^{5} T^{13} - 108 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 15 T + 5 p T^{2} + 3060 T^{3} + 35713 T^{4} + 3060 p T^{5} + 5 p^{3} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 14 T - 9 T^{2} - 147 T^{3} + 6927 T^{4} + 52059 T^{5} + 728307 T^{6} + 4047848 T^{7} - 14238189 T^{8} + 4047848 p T^{9} + 728307 p^{2} T^{10} + 52059 p^{3} T^{11} + 6927 p^{4} T^{12} - 147 p^{5} T^{13} - 9 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 3 T - 2 T^{2} + 345 T^{3} + 220 T^{4} + 100292 T^{5} + 620974 T^{6} - 13580 T^{7} + 31898639 T^{8} - 13580 p T^{9} + 620974 p^{2} T^{10} + 100292 p^{3} T^{11} + 220 p^{4} T^{12} + 345 p^{5} T^{13} - 2 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 9 T - 75 T^{2} - 768 T^{3} + 10031 T^{4} + 24564 T^{5} - 1553178 T^{6} - 1473567 T^{7} + 120297853 T^{8} - 1473567 p T^{9} - 1553178 p^{2} T^{10} + 24564 p^{3} T^{11} + 10031 p^{4} T^{12} - 768 p^{5} T^{13} - 75 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 23 T + 98 T^{2} - 1745 T^{3} - 15470 T^{4} + 98632 T^{5} + 1750974 T^{6} - 1504700 T^{7} - 137136651 T^{8} - 1504700 p T^{9} + 1750974 p^{2} T^{10} + 98632 p^{3} T^{11} - 15470 p^{4} T^{12} - 1745 p^{5} T^{13} + 98 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 - 17 T + 312 T^{2} - 3419 T^{3} + 38939 T^{4} - 3419 p T^{5} + 312 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + 30 T + 361 T^{2} + 2160 T^{3} + 10062 T^{4} + 83370 T^{5} - 577417 T^{6} - 32754150 T^{7} - 458148745 T^{8} - 32754150 p T^{9} - 577417 p^{2} T^{10} + 83370 p^{3} T^{11} + 10062 p^{4} T^{12} + 2160 p^{5} T^{13} + 361 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.61023747686271912805371474434, −4.54024727259950865541387101673, −4.43295677907406097165640776257, −4.28436792698980427956498944415, −4.13136942357903022238739592667, −4.12680709918778909767483507282, −3.81147481718060232861716148673, −3.46935005156872991064665792146, −3.44270425144872427156905038741, −3.38774044461169239021210489690, −3.22100005088968425972415478546, −3.10719803641808313271114866772, −3.05458952335679835689694500721, −2.80154227572982741189479283859, −2.69985767958160594318356433119, −2.59913951390149605061299916573, −2.17286473431979700197527501203, −2.08059041803105408942684592682, −1.98914702687462204010289449255, −1.73441131439206732589481871295, −1.38823752775102821621737059872, −1.26133703331609477442461635164, −1.18022823708267972273262524272, −0.932541764103380128210815072107, −0.53953412143106605936152423069, 0.53953412143106605936152423069, 0.932541764103380128210815072107, 1.18022823708267972273262524272, 1.26133703331609477442461635164, 1.38823752775102821621737059872, 1.73441131439206732589481871295, 1.98914702687462204010289449255, 2.08059041803105408942684592682, 2.17286473431979700197527501203, 2.59913951390149605061299916573, 2.69985767958160594318356433119, 2.80154227572982741189479283859, 3.05458952335679835689694500721, 3.10719803641808313271114866772, 3.22100005088968425972415478546, 3.38774044461169239021210489690, 3.44270425144872427156905038741, 3.46935005156872991064665792146, 3.81147481718060232861716148673, 4.12680709918778909767483507282, 4.13136942357903022238739592667, 4.28436792698980427956498944415, 4.43295677907406097165640776257, 4.54024727259950865541387101673, 4.61023747686271912805371474434
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.