Dirichlet series
| L(s) = 1 | + 5·2-s − 3·3-s + 9·4-s + 15·5-s − 15·6-s + 4·7-s + 5·8-s − 3·9-s + 75·10-s + 5·11-s − 27·12-s + 20·14-s − 45·15-s − 7·16-s − 11·17-s − 15·18-s + 9·19-s + 135·20-s − 12·21-s + 25·22-s − 15·24-s + 102·25-s + 18·27-s + 36·28-s − 12·29-s − 225·30-s + 8·31-s + ⋯ |
| L(s) = 1 | + 3.53·2-s − 1.73·3-s + 9/2·4-s + 6.70·5-s − 6.12·6-s + 1.51·7-s + 1.76·8-s − 9-s + 23.7·10-s + 1.50·11-s − 7.79·12-s + 5.34·14-s − 11.6·15-s − 7/4·16-s − 2.66·17-s − 3.53·18-s + 2.06·19-s + 30.1·20-s − 2.61·21-s + 5.33·22-s − 3.06·24-s + 20.3·25-s + 3.46·27-s + 6.80·28-s − 2.22·29-s − 41.0·30-s + 1.43·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(13^{16} \cdot 31^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.38002\times 10^{12}\) |
| Root analytic conductor: | \(6.46789\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 13^{16} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(118.4185357\) |
| \(L(\frac12)\) | \(\approx\) | \(118.4185357\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 13 | \( 1 \) |
| 31 | \( ( 1 - T )^{8} \) | |
| good | 2 | \( 1 - 5 T + p^{4} T^{2} - 5 p^{3} T^{3} + 11 p^{3} T^{4} - 87 p T^{5} + 155 p T^{6} - 31 p^{4} T^{7} + 731 T^{8} - 31 p^{5} T^{9} + 155 p^{3} T^{10} - 87 p^{4} T^{11} + 11 p^{7} T^{12} - 5 p^{8} T^{13} + p^{10} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 + p T + 4 p T^{2} + p^{3} T^{3} + 67 T^{4} + 124 T^{5} + 235 T^{6} + 44 p^{2} T^{7} + 26 p^{3} T^{8} + 44 p^{3} T^{9} + 235 p^{2} T^{10} + 124 p^{3} T^{11} + 67 p^{4} T^{12} + p^{8} T^{13} + 4 p^{7} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) | |
| 5 | \( 1 - 3 p T + 123 T^{2} - 717 T^{3} + 3289 T^{4} - 498 p^{2} T^{5} + 39947 T^{6} - 22068 p T^{7} + 264523 T^{8} - 22068 p^{2} T^{9} + 39947 p^{2} T^{10} - 498 p^{5} T^{11} + 3289 p^{4} T^{12} - 717 p^{5} T^{13} + 123 p^{6} T^{14} - 3 p^{8} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - 4 T + 36 T^{2} - 108 T^{3} + 578 T^{4} - 1371 T^{5} + 5846 T^{6} - 11754 T^{7} + 45123 T^{8} - 11754 p T^{9} + 5846 p^{2} T^{10} - 1371 p^{3} T^{11} + 578 p^{4} T^{12} - 108 p^{5} T^{13} + 36 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 5 T + 32 T^{2} - 76 T^{3} + 397 T^{4} - 64 p T^{5} + 4685 T^{6} - 6125 T^{7} + 48318 T^{8} - 6125 p T^{9} + 4685 p^{2} T^{10} - 64 p^{4} T^{11} + 397 p^{4} T^{12} - 76 p^{5} T^{13} + 32 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 11 T + 126 T^{2} + 996 T^{3} + 419 p T^{4} + 42792 T^{5} + 234463 T^{6} + 1116927 T^{7} + 4923742 T^{8} + 1116927 p T^{9} + 234463 p^{2} T^{10} + 42792 p^{3} T^{11} + 419 p^{5} T^{12} + 996 p^{5} T^{13} + 126 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 9 T + 106 T^{2} - 708 T^{3} + 4938 T^{4} - 25924 T^{5} + 7442 p T^{6} - 623397 T^{7} + 3004911 T^{8} - 623397 p T^{9} + 7442 p^{3} T^{10} - 25924 p^{3} T^{11} + 4938 p^{4} T^{12} - 708 p^{5} T^{13} + 106 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 98 T^{2} + 84 T^{3} + 4275 T^{4} + 9392 T^{5} + 115707 T^{6} + 424422 T^{7} + 2628074 T^{8} + 424422 p T^{9} + 115707 p^{2} T^{10} + 9392 p^{3} T^{11} + 4275 p^{4} T^{12} + 84 p^{5} T^{13} + 98 p^{6} T^{14} + p^{8} T^{16} \) | |
| 29 | \( 1 + 12 T + 148 T^{2} + 1103 T^{3} + 8738 T^{4} + 53277 T^{5} + 358535 T^{6} + 1929446 T^{7} + 11586852 T^{8} + 1929446 p T^{9} + 358535 p^{2} T^{10} + 53277 p^{3} T^{11} + 8738 p^{4} T^{12} + 1103 p^{5} T^{13} + 148 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 9 T + 165 T^{2} - 991 T^{3} + 12935 T^{4} - 67343 T^{5} + 740502 T^{6} - 3375839 T^{7} + 31495474 T^{8} - 3375839 p T^{9} + 740502 p^{2} T^{10} - 67343 p^{3} T^{11} + 12935 p^{4} T^{12} - 991 p^{5} T^{13} + 165 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 25 T + 476 T^{2} - 6372 T^{3} + 72959 T^{4} - 693235 T^{5} + 5892153 T^{6} - 43853249 T^{7} + 297848517 T^{8} - 43853249 p T^{9} + 5892153 p^{2} T^{10} - 693235 p^{3} T^{11} + 72959 p^{4} T^{12} - 6372 p^{5} T^{13} + 476 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 7 T + 145 T^{2} - 711 T^{3} + 10108 T^{4} - 44326 T^{5} + 602614 T^{6} - 2834324 T^{7} + 30515756 T^{8} - 2834324 p T^{9} + 602614 p^{2} T^{10} - 44326 p^{3} T^{11} + 10108 p^{4} T^{12} - 711 p^{5} T^{13} + 145 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 17 T + 241 T^{2} - 1981 T^{3} + 14608 T^{4} - 51402 T^{5} + 85212 T^{6} + 2325046 T^{7} - 16365036 T^{8} + 2325046 p T^{9} + 85212 p^{2} T^{10} - 51402 p^{3} T^{11} + 14608 p^{4} T^{12} - 1981 p^{5} T^{13} + 241 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 15 T + 8 p T^{2} + 4950 T^{3} + 77587 T^{4} + 727866 T^{5} + 8123777 T^{6} + 61843695 T^{7} + 535702502 T^{8} + 61843695 p T^{9} + 8123777 p^{2} T^{10} + 727866 p^{3} T^{11} + 77587 p^{4} T^{12} + 4950 p^{5} T^{13} + 8 p^{7} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 15 T + 379 T^{2} - 4031 T^{3} + 62177 T^{4} - 536492 T^{5} + 6357685 T^{6} - 46119098 T^{7} + 447458939 T^{8} - 46119098 p T^{9} + 6357685 p^{2} T^{10} - 536492 p^{3} T^{11} + 62177 p^{4} T^{12} - 4031 p^{5} T^{13} + 379 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 11 T + 294 T^{2} - 3523 T^{3} + 47281 T^{4} - 506350 T^{5} + 5141323 T^{6} - 732462 p T^{7} + 381642698 T^{8} - 732462 p^{2} T^{9} + 5141323 p^{2} T^{10} - 506350 p^{3} T^{11} + 47281 p^{4} T^{12} - 3523 p^{5} T^{13} + 294 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 18 T + 598 T^{2} + 7908 T^{3} + 146860 T^{4} + 22621 p T^{5} + 20041239 T^{6} + 165114013 T^{7} + 1686479924 T^{8} + 165114013 p T^{9} + 20041239 p^{2} T^{10} + 22621 p^{4} T^{11} + 146860 p^{4} T^{12} + 7908 p^{5} T^{13} + 598 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 7 T + 390 T^{2} - 2460 T^{3} + 74246 T^{4} - 409796 T^{5} + 8984378 T^{6} - 42737787 T^{7} + 755659863 T^{8} - 42737787 p T^{9} + 8984378 p^{2} T^{10} - 409796 p^{3} T^{11} + 74246 p^{4} T^{12} - 2460 p^{5} T^{13} + 390 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 24 T + 654 T^{2} + 10155 T^{3} + 163680 T^{4} + 1907249 T^{5} + 22776963 T^{6} + 212473704 T^{7} + 2032799812 T^{8} + 212473704 p T^{9} + 22776963 p^{2} T^{10} + 1907249 p^{3} T^{11} + 163680 p^{4} T^{12} + 10155 p^{5} T^{13} + 654 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 33 T + 797 T^{2} - 13771 T^{3} + 206359 T^{4} - 2593505 T^{5} + 29672938 T^{6} - 299405543 T^{7} + 35570814 p T^{8} - 299405543 p T^{9} + 29672938 p^{2} T^{10} - 2593505 p^{3} T^{11} + 206359 p^{4} T^{12} - 13771 p^{5} T^{13} + 797 p^{6} T^{14} - 33 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 13 T + 473 T^{2} - 5847 T^{3} + 113012 T^{4} - 1228906 T^{5} + 16911750 T^{6} - 156075296 T^{7} + 1700063772 T^{8} - 156075296 p T^{9} + 16911750 p^{2} T^{10} - 1228906 p^{3} T^{11} + 113012 p^{4} T^{12} - 5847 p^{5} T^{13} + 473 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 23 T + 618 T^{2} - 10671 T^{3} + 178919 T^{4} - 2353266 T^{5} + 30161411 T^{6} - 320307670 T^{7} + 3279691286 T^{8} - 320307670 p T^{9} + 30161411 p^{2} T^{10} - 2353266 p^{3} T^{11} + 178919 p^{4} T^{12} - 10671 p^{5} T^{13} + 618 p^{6} T^{14} - 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 17 T + 651 T^{2} - 8770 T^{3} + 194932 T^{4} - 2156514 T^{5} + 35045998 T^{6} - 321297201 T^{7} + 4144181257 T^{8} - 321297201 p T^{9} + 35045998 p^{2} T^{10} - 2156514 p^{3} T^{11} + 194932 p^{4} T^{12} - 8770 p^{5} T^{13} + 651 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.48516828367850570248283856399, −3.26728690504389326199029024324, −3.04882183398943201843524734873, −3.01627867566178025080384813016, −2.87926304406713189672205719172, −2.58010119009353675715960097144, −2.57986008550809545778070645003, −2.56369098812306093016217521039, −2.46321224914753663569872214843, −2.40540213552865227348886870683, −2.33851529976272890207512740911, −2.02160650716422703075699071217, −2.00509872089485316997086882074, −1.98399525592978641470314915726, −1.81852672715530796878528152829, −1.60257965146018032002681698409, −1.59714103315055252151266010419, −1.39545593484006253041662928874, −1.38294664816989062239529322724, −1.10996718150326526856827644136, −1.02123635248196542467181386720, −0.71115430261891299297129372552, −0.61359097514786928968182185732, −0.52343375384400490378002102543, −0.16224687885011793493281995452, 0.16224687885011793493281995452, 0.52343375384400490378002102543, 0.61359097514786928968182185732, 0.71115430261891299297129372552, 1.02123635248196542467181386720, 1.10996718150326526856827644136, 1.38294664816989062239529322724, 1.39545593484006253041662928874, 1.59714103315055252151266010419, 1.60257965146018032002681698409, 1.81852672715530796878528152829, 1.98399525592978641470314915726, 2.00509872089485316997086882074, 2.02160650716422703075699071217, 2.33851529976272890207512740911, 2.40540213552865227348886870683, 2.46321224914753663569872214843, 2.56369098812306093016217521039, 2.57986008550809545778070645003, 2.58010119009353675715960097144, 2.87926304406713189672205719172, 3.01627867566178025080384813016, 3.04882183398943201843524734873, 3.26728690504389326199029024324, 3.48516828367850570248283856399
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.