Dirichlet series
| L(s) = 1 | − 3-s − 4·5-s + 4·9-s + 4·11-s + 9·13-s + 4·15-s + 17-s + 3·19-s + 6·25-s + 27-s + 5·29-s − 20·31-s − 4·33-s − 52·37-s − 9·39-s − 8·41-s + 7·43-s − 16·45-s + 16·47-s − 18·49-s − 51-s + 5·53-s − 16·55-s − 3·57-s + 11·59-s + 12·61-s − 36·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 4/3·9-s + 1.20·11-s + 2.49·13-s + 1.03·15-s + 0.242·17-s + 0.688·19-s + 6/5·25-s + 0.192·27-s + 0.928·29-s − 3.59·31-s − 0.696·33-s − 8.54·37-s − 1.44·39-s − 1.24·41-s + 1.06·43-s − 2.38·45-s + 2.33·47-s − 2.57·49-s − 0.140·51-s + 0.686·53-s − 2.15·55-s − 0.397·57-s + 1.43·59-s + 1.53·61-s − 4.46·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 5^{8} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7185.97\) |
| Root analytic conductor: | \(1.74192\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 5^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6662339076\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6662339076\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T + T^{2} )^{4} \) | |
| 19 | \( 1 - 3 T - 43 T^{2} + 21 T^{3} + 1227 T^{4} + 21 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( 1 + T - p T^{2} - 8 T^{3} + 2 T^{4} + 26 T^{5} + 58 T^{6} - 11 p T^{7} - 215 T^{8} - 11 p^{2} T^{9} + 58 p^{2} T^{10} + 26 p^{3} T^{11} + 2 p^{4} T^{12} - 8 p^{5} T^{13} - p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 9 T^{2} + 11 T^{3} + 12 p T^{4} + 11 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 11 | \( ( 1 - 2 T + 9 T^{2} - 45 T^{3} + 223 T^{4} - 45 p T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 13 | \( 1 - 9 T + 34 T^{2} + 21 T^{3} - 725 T^{4} + 3528 T^{5} - 4796 T^{6} - 2136 p T^{7} + 14572 p T^{8} - 2136 p^{2} T^{9} - 4796 p^{2} T^{10} + 3528 p^{3} T^{11} - 725 p^{4} T^{12} + 21 p^{5} T^{13} + 34 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - T - 33 T^{2} + 86 T^{3} + 326 T^{4} - 1520 T^{5} - 3948 T^{6} + 8421 T^{7} + 123921 T^{8} + 8421 p T^{9} - 3948 p^{2} T^{10} - 1520 p^{3} T^{11} + 326 p^{4} T^{12} + 86 p^{5} T^{13} - 33 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 53 T^{2} + 198 T^{3} + 1483 T^{4} - 7524 T^{5} - 4403 T^{6} + 110088 T^{7} - 232949 T^{8} + 110088 p T^{9} - 4403 p^{2} T^{10} - 7524 p^{3} T^{11} + 1483 p^{4} T^{12} + 198 p^{5} T^{13} - 53 p^{6} T^{14} + p^{8} T^{16} \) | |
| 29 | \( 1 - 5 T - 32 T^{2} - 51 T^{3} + 1313 T^{4} + 3448 T^{5} + 15591 T^{6} - 132895 T^{7} - 792020 T^{8} - 132895 p T^{9} + 15591 p^{2} T^{10} + 3448 p^{3} T^{11} + 1313 p^{4} T^{12} - 51 p^{5} T^{13} - 32 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( ( 1 + 10 T + 95 T^{2} + 627 T^{3} + 3691 T^{4} + 627 p T^{5} + 95 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 26 T + 365 T^{2} + 3495 T^{3} + 24686 T^{4} + 3495 p T^{5} + 365 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 + 8 T - 105 T^{2} - 550 T^{3} + 10439 T^{4} + 31486 T^{5} - 616215 T^{6} - 356064 T^{7} + 31448841 T^{8} - 356064 p T^{9} - 616215 p^{2} T^{10} + 31486 p^{3} T^{11} + 10439 p^{4} T^{12} - 550 p^{5} T^{13} - 105 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 7 T - 91 T^{2} + 472 T^{3} + 6354 T^{4} - 18562 T^{5} - 316690 T^{6} + 445407 T^{7} + 12119113 T^{8} + 445407 p T^{9} - 316690 p^{2} T^{10} - 18562 p^{3} T^{11} + 6354 p^{4} T^{12} + 472 p^{5} T^{13} - 91 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 16 T + 93 T^{2} - 70 T^{3} - 3463 T^{4} + 44806 T^{5} - 209457 T^{6} - 6882 p T^{7} + 8533791 T^{8} - 6882 p^{2} T^{9} - 209457 p^{2} T^{10} + 44806 p^{3} T^{11} - 3463 p^{4} T^{12} - 70 p^{5} T^{13} + 93 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 5 T - 69 T^{2} - 86 T^{3} + 2312 T^{4} + 27416 T^{5} - 8940 T^{6} - 1219815 T^{7} + 3737349 T^{8} - 1219815 p T^{9} - 8940 p^{2} T^{10} + 27416 p^{3} T^{11} + 2312 p^{4} T^{12} - 86 p^{5} T^{13} - 69 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 11 T - 60 T^{2} + 979 T^{3} + 443 T^{4} - 4774 T^{5} - 391341 T^{6} - 1207965 T^{7} + 48891330 T^{8} - 1207965 p T^{9} - 391341 p^{2} T^{10} - 4774 p^{3} T^{11} + 443 p^{4} T^{12} + 979 p^{5} T^{13} - 60 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 12 T - 78 T^{2} + 856 T^{3} + 8493 T^{4} - 25976 T^{5} - 905558 T^{6} + 410364 T^{7} + 66133564 T^{8} + 410364 p T^{9} - 905558 p^{2} T^{10} - 25976 p^{3} T^{11} + 8493 p^{4} T^{12} + 856 p^{5} T^{13} - 78 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 144 T^{2} + 112 T^{3} + 7122 T^{4} - 11816 T^{5} - 664448 T^{6} + 318192 T^{7} + 71895859 T^{8} + 318192 p T^{9} - 664448 p^{2} T^{10} - 11816 p^{3} T^{11} + 7122 p^{4} T^{12} + 112 p^{5} T^{13} - 144 p^{6} T^{14} + p^{8} T^{16} \) | |
| 71 | \( 1 - 14 T - 87 T^{2} + 664 T^{3} + 19850 T^{4} - 38806 T^{5} - 1979478 T^{6} + 3533781 T^{7} + 107325432 T^{8} + 3533781 p T^{9} - 1979478 p^{2} T^{10} - 38806 p^{3} T^{11} + 19850 p^{4} T^{12} + 664 p^{5} T^{13} - 87 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 4 T - 139 T^{2} - 1282 T^{3} + 7691 T^{4} + 116588 T^{5} + 336549 T^{6} - 4745052 T^{7} - 52999295 T^{8} - 4745052 p T^{9} + 336549 p^{2} T^{10} + 116588 p^{3} T^{11} + 7691 p^{4} T^{12} - 1282 p^{5} T^{13} - 139 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 13 T - 19 T^{2} + 94 T^{3} + 4641 T^{4} + 30533 T^{5} - 17014 T^{6} - 1995549 T^{7} - 18292886 T^{8} - 1995549 p T^{9} - 17014 p^{2} T^{10} + 30533 p^{3} T^{11} + 4641 p^{4} T^{12} + 94 p^{5} T^{13} - 19 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 - 5 T + 250 T^{2} - 576 T^{3} + 26420 T^{4} - 576 p T^{5} + 250 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 - 5 T - 209 T^{2} + 1332 T^{3} + 21515 T^{4} - 134213 T^{5} - 1599390 T^{6} + 5554751 T^{7} + 127743040 T^{8} + 5554751 p T^{9} - 1599390 p^{2} T^{10} - 134213 p^{3} T^{11} + 21515 p^{4} T^{12} + 1332 p^{5} T^{13} - 209 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + T - 265 T^{2} - 214 T^{3} + 34076 T^{4} + 14240 T^{5} - 4653132 T^{6} - 223893 T^{7} + 569839261 T^{8} - 223893 p T^{9} - 4653132 p^{2} T^{10} + 14240 p^{3} T^{11} + 34076 p^{4} T^{12} - 214 p^{5} T^{13} - 265 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.13996140523067507593566820827, −4.97335775724522582939336293389, −4.89827486425458345349048421789, −4.44402740886320771401966039019, −4.36485583685468051003612318099, −4.20202114297450432123922315205, −4.12147879369052448686279277757, −3.96582561127663072115051724034, −3.77608113204582642130537463465, −3.73248915548215278393516785151, −3.51350988179190147559305669065, −3.42886313477209302230054869207, −3.41937975074693589557824707093, −3.35389435248843869260378582265, −2.99255292093521569468510521869, −2.68729767403263981844063335742, −2.58104200211940855445101081780, −1.96908588031302520275192266506, −1.84213141740888498894496777583, −1.77276247746210895030994729784, −1.75420256348134672667144594515, −1.39795602157506891481188536356, −0.990413314656452807125653544747, −0.900455901628262439171570114433, −0.20270259148807527191702159850, 0.20270259148807527191702159850, 0.900455901628262439171570114433, 0.990413314656452807125653544747, 1.39795602157506891481188536356, 1.75420256348134672667144594515, 1.77276247746210895030994729784, 1.84213141740888498894496777583, 1.96908588031302520275192266506, 2.58104200211940855445101081780, 2.68729767403263981844063335742, 2.99255292093521569468510521869, 3.35389435248843869260378582265, 3.41937975074693589557824707093, 3.42886313477209302230054869207, 3.51350988179190147559305669065, 3.73248915548215278393516785151, 3.77608113204582642130537463465, 3.96582561127663072115051724034, 4.12147879369052448686279277757, 4.20202114297450432123922315205, 4.36485583685468051003612318099, 4.44402740886320771401966039019, 4.89827486425458345349048421789, 4.97335775724522582939336293389, 5.13996140523067507593566820827
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.