Dirichlet series
| L(s) = 1 | − 20·4-s + 22·5-s − 28·11-s + 240·16-s − 42·19-s − 440·20-s + 290·25-s + 216·29-s + 88·31-s + 38·41-s + 560·44-s + 989·49-s − 616·55-s − 2.05e3·59-s + 1.06e3·61-s − 2.24e3·64-s − 2.84e3·71-s + 840·76-s − 1.60e3·79-s + 5.28e3·80-s + 1.13e3·89-s − 924·95-s − 5.80e3·100-s + 4.19e3·101-s − 1.74e3·109-s − 4.32e3·116-s − 3.77e3·121-s + ⋯ |
| L(s) = 1 | − 5/2·4-s + 1.96·5-s − 0.767·11-s + 15/4·16-s − 0.507·19-s − 4.91·20-s + 2.31·25-s + 1.38·29-s + 0.509·31-s + 0.144·41-s + 1.91·44-s + 2.88·49-s − 1.51·55-s − 4.52·59-s + 2.23·61-s − 4.37·64-s − 4.75·71-s + 1.26·76-s − 2.29·79-s + 7.37·80-s + 1.34·89-s − 0.997·95-s − 5.79·100-s + 4.12·101-s − 1.53·109-s − 3.45·116-s − 2.83·121-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{40} \cdot 5^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{40} \cdot 5^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{40} \cdot 5^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.21599\times 10^{16}\) |
| Root analytic conductor: | \(6.91314\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{40} \cdot 5^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(37.96112913\) |
| \(L(\frac12)\) | \(\approx\) | \(37.96112913\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{5} \) |
| 3 | \( 1 \) | |
| 5 | \( 1 - 22 T + 194 T^{2} - 1968 T^{3} + 4801 p T^{4} - 9956 p^{2} T^{5} + 4801 p^{4} T^{6} - 1968 p^{6} T^{7} + 194 p^{9} T^{8} - 22 p^{12} T^{9} + p^{15} T^{10} \) | |
| good | 7 | \( 1 - 989 T^{2} + 11253 p^{2} T^{4} - 216725100 T^{6} + 82684409322 T^{8} - 597847827342 p^{2} T^{10} + 82684409322 p^{6} T^{12} - 216725100 p^{12} T^{14} + 11253 p^{20} T^{16} - 989 p^{24} T^{18} + p^{30} T^{20} \) |
| 11 | \( ( 1 + 14 T + 2180 T^{2} + 45636 T^{3} + 4259435 T^{4} + 28129324 T^{5} + 4259435 p^{3} T^{6} + 45636 p^{6} T^{7} + 2180 p^{9} T^{8} + 14 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 13 | \( 1 - 8467 T^{2} + 41346012 T^{4} - 11961676089 p T^{6} + 462940014645519 T^{8} - 1116082425021637272 T^{10} + 462940014645519 p^{6} T^{12} - 11961676089 p^{13} T^{14} + 41346012 p^{18} T^{16} - 8467 p^{24} T^{18} + p^{30} T^{20} \) | |
| 17 | \( 1 - 27564 T^{2} + 387966374 T^{4} - 215798168622 p T^{6} + 25966368140009113 T^{8} - \)\(14\!\cdots\!04\)\( T^{10} + 25966368140009113 p^{6} T^{12} - 215798168622 p^{13} T^{14} + 387966374 p^{18} T^{16} - 27564 p^{24} T^{18} + p^{30} T^{20} \) | |
| 19 | \( ( 1 + 21 T + 14570 T^{2} + 323139 T^{3} + 157538305 T^{4} + 2409635700 T^{5} + 157538305 p^{3} T^{6} + 323139 p^{6} T^{7} + 14570 p^{9} T^{8} + 21 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 23 | \( 1 - 73005 T^{2} + 2524256189 T^{4} - 56433033194028 T^{6} + 941157412042844530 T^{8} - \)\(12\!\cdots\!54\)\( T^{10} + 941157412042844530 p^{6} T^{12} - 56433033194028 p^{12} T^{14} + 2524256189 p^{18} T^{16} - 73005 p^{24} T^{18} + p^{30} T^{20} \) | |
| 29 | \( ( 1 - 108 T + 93091 T^{2} - 7346268 T^{3} + 3874520917 T^{4} - 237257418168 T^{5} + 3874520917 p^{3} T^{6} - 7346268 p^{6} T^{7} + 93091 p^{9} T^{8} - 108 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 31 | \( ( 1 - 44 T + 87240 T^{2} + 2084718 T^{3} + 3208376163 T^{4} + 199557551580 T^{5} + 3208376163 p^{3} T^{6} + 2084718 p^{6} T^{7} + 87240 p^{9} T^{8} - 44 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 37 | \( 1 - 436280 T^{2} + 88749670398 T^{4} - 11064509169435714 T^{6} + \)\(93\!\cdots\!21\)\( T^{8} - \)\(55\!\cdots\!92\)\( T^{10} + \)\(93\!\cdots\!21\)\( p^{6} T^{12} - 11064509169435714 p^{12} T^{14} + 88749670398 p^{18} T^{16} - 436280 p^{24} T^{18} + p^{30} T^{20} \) | |
| 41 | \( ( 1 - 19 T + 104036 T^{2} + 27771483 T^{3} + 5625867263 T^{4} + 2838087609352 T^{5} + 5625867263 p^{3} T^{6} + 27771483 p^{6} T^{7} + 104036 p^{9} T^{8} - 19 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 43 | \( 1 - 547898 T^{2} + 142594310133 T^{4} - 23619463504397688 T^{6} + \)\(28\!\cdots\!54\)\( T^{8} - \)\(25\!\cdots\!88\)\( T^{10} + \)\(28\!\cdots\!54\)\( p^{6} T^{12} - 23619463504397688 p^{12} T^{14} + 142594310133 p^{18} T^{16} - 547898 p^{24} T^{18} + p^{30} T^{20} \) | |
| 47 | \( 1 - 160917 T^{2} + 32408020925 T^{4} - 3943665433054572 T^{6} + \)\(59\!\cdots\!70\)\( T^{8} - \)\(57\!\cdots\!82\)\( T^{10} + \)\(59\!\cdots\!70\)\( p^{6} T^{12} - 3943665433054572 p^{12} T^{14} + 32408020925 p^{18} T^{16} - 160917 p^{24} T^{18} + p^{30} T^{20} \) | |
| 53 | \( 1 - 481721 T^{2} + 124883352789 T^{4} - 26025554285726556 T^{6} + \)\(48\!\cdots\!38\)\( T^{8} - \)\(78\!\cdots\!26\)\( T^{10} + \)\(48\!\cdots\!38\)\( p^{6} T^{12} - 26025554285726556 p^{12} T^{14} + 124883352789 p^{18} T^{16} - 481721 p^{24} T^{18} + p^{30} T^{20} \) | |
| 59 | \( ( 1 + 1025 T + 934574 T^{2} + 580021995 T^{3} + 352700074433 T^{4} + 162166394208160 T^{5} + 352700074433 p^{3} T^{6} + 580021995 p^{6} T^{7} + 934574 p^{9} T^{8} + 1025 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 61 | \( ( 1 - 532 T + 39126 T^{2} + 81398814 T^{3} + 68239959321 T^{4} - 58390875038148 T^{5} + 68239959321 p^{3} T^{6} + 81398814 p^{6} T^{7} + 39126 p^{9} T^{8} - 532 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 67 | \( 1 - 1586630 T^{2} + 1379172381045 T^{4} - 812854783643216040 T^{6} + \)\(35\!\cdots\!30\)\( T^{8} - \)\(12\!\cdots\!12\)\( T^{10} + \)\(35\!\cdots\!30\)\( p^{6} T^{12} - 812854783643216040 p^{12} T^{14} + 1379172381045 p^{18} T^{16} - 1586630 p^{24} T^{18} + p^{30} T^{20} \) | |
| 71 | \( ( 1 + 1422 T + 1367800 T^{2} + 667635588 T^{3} + 237071942839 T^{4} + 39943322814060 T^{5} + 237071942839 p^{3} T^{6} + 667635588 p^{6} T^{7} + 1367800 p^{9} T^{8} + 1422 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 73 | \( 1 - 1686776 T^{2} + 1611760751694 T^{4} - 1069814961419818746 T^{6} + \)\(55\!\cdots\!93\)\( T^{8} - \)\(23\!\cdots\!96\)\( T^{10} + \)\(55\!\cdots\!93\)\( p^{6} T^{12} - 1069814961419818746 p^{12} T^{14} + 1611760751694 p^{18} T^{16} - 1686776 p^{24} T^{18} + p^{30} T^{20} \) | |
| 79 | \( ( 1 + 804 T + 1138415 T^{2} + 981795360 T^{3} + 809210827198 T^{4} + 716956068535416 T^{5} + 809210827198 p^{3} T^{6} + 981795360 p^{6} T^{7} + 1138415 p^{9} T^{8} + 804 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 83 | \( 1 - 4865814 T^{2} + 11000705361557 T^{4} - 15256377151494541416 T^{6} + \)\(14\!\cdots\!14\)\( T^{8} - \)\(96\!\cdots\!00\)\( T^{10} + \)\(14\!\cdots\!14\)\( p^{6} T^{12} - 15256377151494541416 p^{12} T^{14} + 11000705361557 p^{18} T^{16} - 4865814 p^{24} T^{18} + p^{30} T^{20} \) | |
| 89 | \( ( 1 - 566 T + 1553291 T^{2} + 305869032 T^{3} + 513050516045 T^{4} + 915246283006106 T^{5} + 513050516045 p^{3} T^{6} + 305869032 p^{6} T^{7} + 1553291 p^{9} T^{8} - 566 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 97 | \( 1 - 4862354 T^{2} + 13215632715069 T^{4} - 24078303722833694904 T^{6} + \)\(32\!\cdots\!98\)\( T^{8} - \)\(33\!\cdots\!04\)\( T^{10} + \)\(32\!\cdots\!98\)\( p^{6} T^{12} - 24078303722833694904 p^{12} T^{14} + 13215632715069 p^{18} T^{16} - 4862354 p^{24} T^{18} + p^{30} T^{20} \) | |
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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
−3.20012574991461262710978635113, −3.11716804542729319821410206107, −2.97780153356235720578147589868, −2.96830211297804533488749467709, −2.86220030994256645266021020238, −2.81477574390382032654998240856, −2.73857273264401053546307640316, −2.70423261202175805403604164022, −2.36359121162051749389777895123, −2.21412511758739147103541661629, −2.02889000047207762832615659699, −1.82336654075094747969050728901, −1.73646346877050199827174811808, −1.69622047092740210548819898228, −1.67354250906786515053071774960, −1.53606155759569238903463756194, −1.44951430205188373343506235215, −1.11013987304757603036125882511, −0.932431330277150885729181405210, −0.67224859964194900460775949325, −0.62532817631652801605204159546, −0.55404822595746457559784160385, −0.45147084300872390607139243010, −0.39164367490995192215528324954, −0.31907315472650086965856078249, 0.31907315472650086965856078249, 0.39164367490995192215528324954, 0.45147084300872390607139243010, 0.55404822595746457559784160385, 0.62532817631652801605204159546, 0.67224859964194900460775949325, 0.932431330277150885729181405210, 1.11013987304757603036125882511, 1.44951430205188373343506235215, 1.53606155759569238903463756194, 1.67354250906786515053071774960, 1.69622047092740210548819898228, 1.73646346877050199827174811808, 1.82336654075094747969050728901, 2.02889000047207762832615659699, 2.21412511758739147103541661629, 2.36359121162051749389777895123, 2.70423261202175805403604164022, 2.73857273264401053546307640316, 2.81477574390382032654998240856, 2.86220030994256645266021020238, 2.96830211297804533488749467709, 2.97780153356235720578147589868, 3.11716804542729319821410206107, 3.20012574991461262710978635113
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.