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 − 8·13-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 + 40·26-s + 18·27-s − 36·28-s − 12·29-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 − 2.21·13-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 + 7.84·26-s + 3.46·27-s − 6.80·28-s − 2.22·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(13^{8} \cdot 31^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(11498.9\) |
| Root analytic conductor: | \(1.79387\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 13^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | 13 | \( ( 1 + T )^{8} \) |
| 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
−5.55919912917070163629223264068, −5.17982604372843120280706097640, −5.13691829535945396119705295135, −5.08963879308463555324045935724, −5.04454236254919929545568131099, −4.98965437039789436900893820291, −4.74933227691358777173825891207, −4.45502047937659590282333282345, −4.32818034571112468288800247797, −4.31367599863913343511589802024, −4.04020528279066032307437325403, −3.81528894761928973337490945168, −3.76510791335226692506830073647, −3.69473040411012891935736439119, −3.59812873379145027428298674004, −3.53484852311102144477275489243, −3.14247265535570356753816606078, −3.02480756303301781897269614583, −2.79025995339998899099059676332, −2.74397615181231491808909336759, −2.31023457792955076016158407074, −2.14198710893907457352447783020, −2.06326340777272610218749326388, −1.60986859223855936996153615685, −1.14071077133665619971705480402, 0, 0, 0, 0, 0, 0, 0, 0, 1.14071077133665619971705480402, 1.60986859223855936996153615685, 2.06326340777272610218749326388, 2.14198710893907457352447783020, 2.31023457792955076016158407074, 2.74397615181231491808909336759, 2.79025995339998899099059676332, 3.02480756303301781897269614583, 3.14247265535570356753816606078, 3.53484852311102144477275489243, 3.59812873379145027428298674004, 3.69473040411012891935736439119, 3.76510791335226692506830073647, 3.81528894761928973337490945168, 4.04020528279066032307437325403, 4.31367599863913343511589802024, 4.32818034571112468288800247797, 4.45502047937659590282333282345, 4.74933227691358777173825891207, 4.98965437039789436900893820291, 5.04454236254919929545568131099, 5.08963879308463555324045935724, 5.13691829535945396119705295135, 5.17982604372843120280706097640, 5.55919912917070163629223264068
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.