Dirichlet series
| L(s) = 1 | + 97·9-s + 8·11-s − 190·19-s − 134·29-s + 876·31-s + 1.22e3·41-s + 965·49-s − 1.08e3·59-s + 688·61-s + 440·71-s − 4.57e3·79-s + 4.68e3·81-s + 2.62e3·89-s + 776·99-s − 1.85e3·101-s − 2.10e3·109-s − 8.84e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.27e4·169-s + ⋯ |
| L(s) = 1 | + 3.59·9-s + 0.219·11-s − 2.29·19-s − 0.858·29-s + 5.07·31-s + 4.64·41-s + 2.81·49-s − 2.38·59-s + 1.44·61-s + 0.735·71-s − 6.51·79-s + 6.41·81-s + 3.12·89-s + 0.787·99-s − 1.82·101-s − 1.85·109-s − 6.64·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 5.80·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{20} \cdot 5^{20} \cdot 19^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.13470\times 10^{20}\) |
| Root analytic conductor: | \(10.5879\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{20} \cdot 5^{20} \cdot 19^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.1090564446\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1090564446\) |
| \(L(\frac{5}{2})\) | 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 | \( 1 \) | |
| 19 | \( ( 1 + p T )^{10} \) | |
| good | 3 | \( 1 - 97 T^{2} + 4729 T^{4} - 164348 T^{6} + 4650158 T^{8} - 122536886 T^{10} + 4650158 p^{6} T^{12} - 164348 p^{12} T^{14} + 4729 p^{18} T^{16} - 97 p^{24} T^{18} + p^{30} T^{20} \) |
| 7 | \( 1 - 965 T^{2} + 452181 T^{4} - 164681676 T^{6} + 56573476602 T^{8} - 19195116386718 T^{10} + 56573476602 p^{6} T^{12} - 164681676 p^{12} T^{14} + 452181 p^{18} T^{16} - 965 p^{24} T^{18} + p^{30} T^{20} \) | |
| 11 | \( ( 1 - 4 T + 4447 T^{2} + 8560 T^{3} + 9328618 T^{4} + 35711592 T^{5} + 9328618 p^{3} T^{6} + 8560 p^{6} T^{7} + 4447 p^{9} T^{8} - 4 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 13 | \( 1 - 12749 T^{2} + 83841369 T^{4} - 369774896620 T^{6} + 1202198951458862 T^{8} - 2999254962563894334 T^{10} + 1202198951458862 p^{6} T^{12} - 369774896620 p^{12} T^{14} + 83841369 p^{18} T^{16} - 12749 p^{24} T^{18} + p^{30} T^{20} \) | |
| 17 | \( 1 - 15065 T^{2} + 124482261 T^{4} - 669360946460 T^{6} + 2557357522023370 T^{8} - 10509037032484305174 T^{10} + 2557357522023370 p^{6} T^{12} - 669360946460 p^{12} T^{14} + 124482261 p^{18} T^{16} - 15065 p^{24} T^{18} + p^{30} T^{20} \) | |
| 23 | \( 1 - 55373 T^{2} + 1834841301 T^{4} - 42372965158316 T^{6} + 742344321188765722 T^{8} - \)\(10\!\cdots\!98\)\( T^{10} + 742344321188765722 p^{6} T^{12} - 42372965158316 p^{12} T^{14} + 1834841301 p^{18} T^{16} - 55373 p^{24} T^{18} + p^{30} T^{20} \) | |
| 29 | \( ( 1 + 67 T + 49229 T^{2} + 5483516 T^{3} + 756138406 T^{4} + 187228716498 T^{5} + 756138406 p^{3} T^{6} + 5483516 p^{6} T^{7} + 49229 p^{9} T^{8} + 67 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 31 | \( ( 1 - 438 T + 180243 T^{2} - 47055624 T^{3} + 11357594210 T^{4} - 2031936242820 T^{5} + 11357594210 p^{3} T^{6} - 47055624 p^{6} T^{7} + 180243 p^{9} T^{8} - 438 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 37 | \( 1 - 153290 T^{2} + 9711851949 T^{4} - 136351338451560 T^{6} - 23202520219290401814 T^{8} + \)\(19\!\cdots\!08\)\( T^{10} - 23202520219290401814 p^{6} T^{12} - 136351338451560 p^{12} T^{14} + 9711851949 p^{18} T^{16} - 153290 p^{24} T^{18} + p^{30} T^{20} \) | |
| 41 | \( ( 1 - 610 T + 396661 T^{2} - 163710296 T^{3} + 58238228098 T^{4} - 16771094841804 T^{5} + 58238228098 p^{3} T^{6} - 163710296 p^{6} T^{7} + 396661 p^{9} T^{8} - 610 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 43 | \( 1 - 162666 T^{2} + 15947645845 T^{4} - 1341147716651128 T^{6} + \)\(15\!\cdots\!06\)\( T^{8} - \)\(15\!\cdots\!68\)\( T^{10} + \)\(15\!\cdots\!06\)\( p^{6} T^{12} - 1341147716651128 p^{12} T^{14} + 15947645845 p^{18} T^{16} - 162666 p^{24} T^{18} + p^{30} T^{20} \) | |
| 47 | \( 1 - 489766 T^{2} + 108483126221 T^{4} - 14247387977171528 T^{6} + \)\(13\!\cdots\!06\)\( T^{8} - \)\(12\!\cdots\!64\)\( T^{10} + \)\(13\!\cdots\!06\)\( p^{6} T^{12} - 14247387977171528 p^{12} T^{14} + 108483126221 p^{18} T^{16} - 489766 p^{24} T^{18} + p^{30} T^{20} \) | |
| 53 | \( 1 - 607157 T^{2} + 189903540153 T^{4} - 40372292507854316 T^{6} + \)\(68\!\cdots\!38\)\( T^{8} - \)\(10\!\cdots\!38\)\( T^{10} + \)\(68\!\cdots\!38\)\( p^{6} T^{12} - 40372292507854316 p^{12} T^{14} + 189903540153 p^{18} T^{16} - 607157 p^{24} T^{18} + p^{30} T^{20} \) | |
| 59 | \( ( 1 + 541 T + 332383 T^{2} + 51303836 T^{3} + 67819177738 T^{4} + 20116641719262 T^{5} + 67819177738 p^{3} T^{6} + 51303836 p^{6} T^{7} + 332383 p^{9} T^{8} + 541 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 61 | \( ( 1 - 344 T + 815073 T^{2} - 297780688 T^{3} + 5041921562 p T^{4} - 99740628719472 T^{5} + 5041921562 p^{4} T^{6} - 297780688 p^{6} T^{7} + 815073 p^{9} T^{8} - 344 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 67 | \( 1 - 884369 T^{2} + 592014904473 T^{4} - 287204698747920700 T^{6} + \)\(11\!\cdots\!34\)\( T^{8} - \)\(37\!\cdots\!50\)\( T^{10} + \)\(11\!\cdots\!34\)\( p^{6} T^{12} - 287204698747920700 p^{12} T^{14} + 592014904473 p^{18} T^{16} - 884369 p^{24} T^{18} + p^{30} T^{20} \) | |
| 71 | \( ( 1 - 220 T + 1163187 T^{2} - 458835856 T^{3} + 598823745130 T^{4} - 272272347172520 T^{5} + 598823745130 p^{3} T^{6} - 458835856 p^{6} T^{7} + 1163187 p^{9} T^{8} - 220 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 73 | \( 1 - 2676857 T^{2} + 3376527585573 T^{4} - 2686470045402171612 T^{6} + \)\(15\!\cdots\!14\)\( T^{8} - \)\(67\!\cdots\!94\)\( T^{10} + \)\(15\!\cdots\!14\)\( p^{6} T^{12} - 2686470045402171612 p^{12} T^{14} + 3376527585573 p^{18} T^{16} - 2676857 p^{24} T^{18} + p^{30} T^{20} \) | |
| 79 | \( ( 1 + 2286 T + 3956395 T^{2} + 4739351720 T^{3} + 4618343076682 T^{4} + 3558112335635156 T^{5} + 4618343076682 p^{3} T^{6} + 4739351720 p^{6} T^{7} + 3956395 p^{9} T^{8} + 2286 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 83 | \( 1 - 2598938 T^{2} + 3875292412581 T^{4} - 4122110172858073016 T^{6} + \)\(33\!\cdots\!82\)\( T^{8} - \)\(21\!\cdots\!88\)\( T^{10} + \)\(33\!\cdots\!82\)\( p^{6} T^{12} - 4122110172858073016 p^{12} T^{14} + 3875292412581 p^{18} T^{16} - 2598938 p^{24} T^{18} + p^{30} T^{20} \) | |
| 89 | \( ( 1 - 1310 T + 1833397 T^{2} - 115232680 T^{3} - 454504481198 T^{4} + 1404342874829388 T^{5} - 454504481198 p^{3} T^{6} - 115232680 p^{6} T^{7} + 1833397 p^{9} T^{8} - 1310 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 97 | \( 1 - 6880322 T^{2} + 229539686277 p T^{4} - 45246729120843487752 T^{6} + \)\(64\!\cdots\!26\)\( T^{8} - \)\(68\!\cdots\!84\)\( T^{10} + \)\(64\!\cdots\!26\)\( p^{6} T^{12} - 45246729120843487752 p^{12} T^{14} + 229539686277 p^{19} T^{16} - 6880322 p^{24} T^{18} + p^{30} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.82991509025072895617553702576, −2.65952485502998142802973293893, −2.64617648449383699736194014716, −2.61400820238370781412898394782, −2.44395505525193995122818922313, −2.28820692563881392859102161431, −2.23405840879181017951500940601, −2.16827425089326198400222730124, −2.13009851568507647600245149533, −2.05439000985209369001072970248, −1.68087451655501282222998126758, −1.57772338721805703403096406413, −1.48318600099407123281576413023, −1.35881697664962269755820334815, −1.27474931045972210068976072990, −1.26972952402033251201824322630, −1.10420348503545236004057240920, −1.02791114678030163673935754754, −0.999574598251685226405643972663, −0.860512378080500543039250866334, −0.63120920546005934951282526146, −0.46613495492537734954414159477, −0.43118506871039187451292841932, −0.082257819260734454519958181223, −0.02219705336291442010646157242, 0.02219705336291442010646157242, 0.082257819260734454519958181223, 0.43118506871039187451292841932, 0.46613495492537734954414159477, 0.63120920546005934951282526146, 0.860512378080500543039250866334, 0.999574598251685226405643972663, 1.02791114678030163673935754754, 1.10420348503545236004057240920, 1.26972952402033251201824322630, 1.27474931045972210068976072990, 1.35881697664962269755820334815, 1.48318600099407123281576413023, 1.57772338721805703403096406413, 1.68087451655501282222998126758, 2.05439000985209369001072970248, 2.13009851568507647600245149533, 2.16827425089326198400222730124, 2.23405840879181017951500940601, 2.28820692563881392859102161431, 2.44395505525193995122818922313, 2.61400820238370781412898394782, 2.64617648449383699736194014716, 2.65952485502998142802973293893, 2.82991509025072895617553702576
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.