Dirichlet series
| L(s) = 1 | − 110·2-s − 660·3-s + 4.82e3·4-s + 7.26e4·6-s − 7.83e4·8-s − 1.59e6·9-s − 4.59e6·11-s − 3.18e6·12-s − 1.11e7·16-s + 4.28e7·17-s + 1.75e8·18-s + 3.60e7·19-s + 5.05e8·22-s + 5.17e7·24-s + 8.67e8·25-s + 1.72e9·27-s + 6.21e8·32-s + 3.03e9·33-s − 4.71e9·34-s − 7.69e9·36-s − 3.96e9·38-s − 1.23e10·41-s + 2.50e10·43-s − 2.21e10·44-s + 7.33e9·48-s + 4.94e10·49-s − 9.53e10·50-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 0.905·3-s + 1.17·4-s + 1.55·6-s − 0.298·8-s − 2.99·9-s − 2.59·11-s − 1.06·12-s − 0.662·16-s + 1.77·17-s + 5.15·18-s + 0.766·19-s + 4.45·22-s + 0.270·24-s + 3.55·25-s + 4.45·27-s + 0.578·32-s + 2.34·33-s − 3.05·34-s − 3.53·36-s − 1.31·38-s − 2.59·41-s + 3.96·43-s − 3.05·44-s + 0.599·48-s + 3.57·49-s − 6.10·50-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(13-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30}\right)^{s/2} \, \Gamma_{\C}(s+6)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{30}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.36850\times 10^{8}\) |
| Root analytic conductor: | \(2.70406\) |
| Motivic weight: | \(12\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{30} ,\ ( \ : [6]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{13}{2})\) | \(\approx\) | \(0.04033700415\) |
| \(L(\frac12)\) | \(\approx\) | \(0.04033700415\) |
| \(L(7)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + 55 p T + 909 p^{3} T^{2} + 5425 p^{6} T^{3} + 22283 p^{10} T^{4} + 61185 p^{15} T^{5} + 22283 p^{22} T^{6} + 5425 p^{30} T^{7} + 909 p^{39} T^{8} + 55 p^{49} T^{9} + p^{60} T^{10} \) |
| good | 3 | \( ( 1 + 110 p T + 319951 p T^{2} + 65480 p^{5} T^{3} + 609547090 p^{6} T^{4} - 8988906860 p^{9} T^{5} + 609547090 p^{18} T^{6} + 65480 p^{29} T^{7} + 319951 p^{37} T^{8} + 110 p^{49} T^{9} + p^{60} T^{10} )^{2} \) |
| 5 | \( 1 - 173445986 p T^{2} + 94306587583078569 p T^{4} - \)\(15\!\cdots\!04\)\( p^{3} T^{6} + \)\(80\!\cdots\!02\)\( p^{7} T^{8} - \)\(17\!\cdots\!76\)\( p^{10} T^{10} + \)\(80\!\cdots\!02\)\( p^{31} T^{12} - \)\(15\!\cdots\!04\)\( p^{51} T^{14} + 94306587583078569 p^{73} T^{16} - 173445986 p^{97} T^{18} + p^{120} T^{20} \) | |
| 7 | \( 1 - 49440192586 T^{2} + \)\(16\!\cdots\!33\)\( T^{4} - \)\(52\!\cdots\!24\)\( p T^{6} + \)\(20\!\cdots\!82\)\( p^{3} T^{8} - \)\(61\!\cdots\!16\)\( p^{5} T^{10} + \)\(20\!\cdots\!82\)\( p^{27} T^{12} - \)\(52\!\cdots\!24\)\( p^{49} T^{14} + \)\(16\!\cdots\!33\)\( p^{72} T^{16} - 49440192586 p^{96} T^{18} + p^{120} T^{20} \) | |
| 11 | \( ( 1 + 208702 p T + 7492784703 p^{3} T^{2} + 10162510546892360 p^{3} T^{3} + \)\(25\!\cdots\!42\)\( p^{4} T^{4} + \)\(24\!\cdots\!64\)\( p^{5} T^{5} + \)\(25\!\cdots\!42\)\( p^{16} T^{6} + 10162510546892360 p^{27} T^{7} + 7492784703 p^{39} T^{8} + 208702 p^{49} T^{9} + p^{60} T^{10} )^{2} \) | |
| 13 | \( 1 - 104531244325546 T^{2} + \)\(58\!\cdots\!13\)\( T^{4} - \)\(21\!\cdots\!88\)\( T^{6} + \)\(61\!\cdots\!26\)\( T^{8} - \)\(14\!\cdots\!12\)\( T^{10} + \)\(61\!\cdots\!26\)\( p^{24} T^{12} - \)\(21\!\cdots\!88\)\( p^{48} T^{14} + \)\(58\!\cdots\!13\)\( p^{72} T^{16} - 104531244325546 p^{96} T^{18} + p^{120} T^{20} \) | |
| 17 | \( ( 1 - 21438250 T + 2412850471912173 T^{2} - \)\(42\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!10\)\( T^{4} - \)\(35\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!10\)\( p^{12} T^{6} - \)\(42\!\cdots\!00\)\( p^{24} T^{7} + 2412850471912173 p^{36} T^{8} - 21438250 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 19 | \( ( 1 - 18031414 T + 8471849079978669 T^{2} - \)\(17\!\cdots\!80\)\( T^{3} + \)\(32\!\cdots\!66\)\( T^{4} - \)\(59\!\cdots\!60\)\( T^{5} + \)\(32\!\cdots\!66\)\( p^{12} T^{6} - \)\(17\!\cdots\!80\)\( p^{24} T^{7} + 8471849079978669 p^{36} T^{8} - 18031414 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 23 | \( 1 - 111670895161565386 T^{2} + \)\(29\!\cdots\!31\)\( p T^{4} - \)\(27\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!94\)\( p^{2} T^{8} - \)\(76\!\cdots\!92\)\( p^{4} T^{10} + \)\(16\!\cdots\!94\)\( p^{26} T^{12} - \)\(27\!\cdots\!68\)\( p^{48} T^{14} + \)\(29\!\cdots\!31\)\( p^{73} T^{16} - 111670895161565386 p^{96} T^{18} + p^{120} T^{20} \) | |
| 29 | \( 1 - 2227945431622225450 T^{2} + \)\(25\!\cdots\!05\)\( T^{4} - \)\(18\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!50\)\( T^{8} - \)\(41\!\cdots\!52\)\( T^{10} + \)\(10\!\cdots\!50\)\( p^{24} T^{12} - \)\(18\!\cdots\!80\)\( p^{48} T^{14} + \)\(25\!\cdots\!05\)\( p^{72} T^{16} - 2227945431622225450 p^{96} T^{18} + p^{120} T^{20} \) | |
| 31 | \( 1 - 4692181618679558410 T^{2} + \)\(10\!\cdots\!45\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!10\)\( p^{24} T^{12} - \)\(13\!\cdots\!20\)\( p^{48} T^{14} + \)\(10\!\cdots\!45\)\( p^{72} T^{16} - 4692181618679558410 p^{96} T^{18} + p^{120} T^{20} \) | |
| 37 | \( 1 - 9412964374123010026 T^{2} + \)\(12\!\cdots\!53\)\( T^{4} - \)\(23\!\cdots\!28\)\( T^{6} + \)\(16\!\cdots\!06\)\( T^{8} + \)\(22\!\cdots\!88\)\( T^{10} + \)\(16\!\cdots\!06\)\( p^{24} T^{12} - \)\(23\!\cdots\!28\)\( p^{48} T^{14} + \)\(12\!\cdots\!53\)\( p^{72} T^{16} - 9412964374123010026 p^{96} T^{18} + p^{120} T^{20} \) | |
| 41 | \( ( 1 + 6169624022 T + 92150338791973341453 T^{2} + \)\(36\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!42\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(33\!\cdots\!42\)\( p^{12} T^{6} + \)\(36\!\cdots\!80\)\( p^{24} T^{7} + 92150338791973341453 p^{36} T^{8} + 6169624022 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 43 | \( ( 1 - 12540747670 T + \)\(16\!\cdots\!37\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!42\)\( T^{4} - \)\(69\!\cdots\!60\)\( T^{5} + \)\(10\!\cdots\!42\)\( p^{12} T^{6} - \)\(13\!\cdots\!00\)\( p^{24} T^{7} + \)\(16\!\cdots\!37\)\( p^{36} T^{8} - 12540747670 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 47 | \( 1 - \)\(58\!\cdots\!66\)\( T^{2} + \)\(14\!\cdots\!13\)\( T^{4} - \)\(16\!\cdots\!68\)\( T^{6} + \)\(59\!\cdots\!26\)\( T^{8} + \)\(35\!\cdots\!88\)\( T^{10} + \)\(59\!\cdots\!26\)\( p^{24} T^{12} - \)\(16\!\cdots\!68\)\( p^{48} T^{14} + \)\(14\!\cdots\!13\)\( p^{72} T^{16} - \)\(58\!\cdots\!66\)\( p^{96} T^{18} + p^{120} T^{20} \) | |
| 53 | \( 1 - \)\(18\!\cdots\!46\)\( T^{2} + \)\(17\!\cdots\!53\)\( T^{4} - \)\(11\!\cdots\!08\)\( T^{6} + \)\(69\!\cdots\!06\)\( T^{8} - \)\(36\!\cdots\!12\)\( T^{10} + \)\(69\!\cdots\!06\)\( p^{24} T^{12} - \)\(11\!\cdots\!08\)\( p^{48} T^{14} + \)\(17\!\cdots\!53\)\( p^{72} T^{16} - \)\(18\!\cdots\!46\)\( p^{96} T^{18} + p^{120} T^{20} \) | |
| 59 | \( ( 1 + 54724013354 T + \)\(77\!\cdots\!53\)\( T^{2} + \)\(32\!\cdots\!72\)\( T^{3} + \)\(25\!\cdots\!86\)\( T^{4} + \)\(13\!\cdots\!52\)\( p T^{5} + \)\(25\!\cdots\!86\)\( p^{12} T^{6} + \)\(32\!\cdots\!72\)\( p^{24} T^{7} + \)\(77\!\cdots\!53\)\( p^{36} T^{8} + 54724013354 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 61 | \( 1 - \)\(20\!\cdots\!90\)\( T^{2} + \)\(20\!\cdots\!25\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{6} + \)\(55\!\cdots\!90\)\( T^{8} - \)\(46\!\cdots\!12\)\( p^{2} T^{10} + \)\(55\!\cdots\!90\)\( p^{24} T^{12} - \)\(12\!\cdots\!00\)\( p^{48} T^{14} + \)\(20\!\cdots\!25\)\( p^{72} T^{16} - \)\(20\!\cdots\!90\)\( p^{96} T^{18} + p^{120} T^{20} \) | |
| 67 | \( ( 1 + 24930448010 T + \)\(26\!\cdots\!33\)\( T^{2} + \)\(32\!\cdots\!80\)\( T^{3} + \)\(51\!\cdots\!30\)\( p T^{4} + \)\(25\!\cdots\!40\)\( T^{5} + \)\(51\!\cdots\!30\)\( p^{13} T^{6} + \)\(32\!\cdots\!80\)\( p^{24} T^{7} + \)\(26\!\cdots\!33\)\( p^{36} T^{8} + 24930448010 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 71 | \( 1 - \)\(23\!\cdots\!50\)\( T^{2} + \)\(94\!\cdots\!05\)\( T^{4} - \)\(19\!\cdots\!80\)\( T^{6} + \)\(48\!\cdots\!50\)\( T^{8} - \)\(69\!\cdots\!52\)\( T^{10} + \)\(48\!\cdots\!50\)\( p^{24} T^{12} - \)\(19\!\cdots\!80\)\( p^{48} T^{14} + \)\(94\!\cdots\!05\)\( p^{72} T^{16} - \)\(23\!\cdots\!50\)\( p^{96} T^{18} + p^{120} T^{20} \) | |
| 73 | \( ( 1 + 29417796470 T + \)\(32\!\cdots\!93\)\( T^{2} - \)\(28\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!10\)\( T^{4} - \)\(28\!\cdots\!20\)\( T^{5} + \)\(11\!\cdots\!10\)\( p^{12} T^{6} - \)\(28\!\cdots\!40\)\( p^{24} T^{7} + \)\(32\!\cdots\!93\)\( p^{36} T^{8} + 29417796470 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 79 | \( 1 - \)\(34\!\cdots\!10\)\( T^{2} + \)\(58\!\cdots\!85\)\( T^{4} - \)\(66\!\cdots\!60\)\( T^{6} + \)\(55\!\cdots\!10\)\( T^{8} - \)\(36\!\cdots\!52\)\( T^{10} + \)\(55\!\cdots\!10\)\( p^{24} T^{12} - \)\(66\!\cdots\!60\)\( p^{48} T^{14} + \)\(58\!\cdots\!85\)\( p^{72} T^{16} - \)\(34\!\cdots\!10\)\( p^{96} T^{18} + p^{120} T^{20} \) | |
| 83 | \( ( 1 + 73118488970 T + \)\(44\!\cdots\!13\)\( T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(84\!\cdots\!10\)\( T^{4} + \)\(20\!\cdots\!80\)\( T^{5} + \)\(84\!\cdots\!10\)\( p^{12} T^{6} + \)\(17\!\cdots\!60\)\( p^{24} T^{7} + \)\(44\!\cdots\!13\)\( p^{36} T^{8} + 73118488970 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 89 | \( ( 1 - 456670757194 T + \)\(80\!\cdots\!49\)\( T^{2} - \)\(29\!\cdots\!80\)\( T^{3} + \)\(34\!\cdots\!26\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{5} + \)\(34\!\cdots\!26\)\( p^{12} T^{6} - \)\(29\!\cdots\!80\)\( p^{24} T^{7} + \)\(80\!\cdots\!49\)\( p^{36} T^{8} - 456670757194 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 97 | \( ( 1 + 737113220630 T + \)\(12\!\cdots\!77\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} - \)\(27\!\cdots\!58\)\( T^{4} - \)\(11\!\cdots\!60\)\( T^{5} - \)\(27\!\cdots\!58\)\( p^{12} T^{6} - \)\(35\!\cdots\!00\)\( p^{24} T^{7} + \)\(12\!\cdots\!77\)\( p^{36} T^{8} + 737113220630 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
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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
−6.55563985544091262671822436038, −6.51067907453677788855868915565, −6.50632904210740071021581079855, −5.68848182162305454268147159134, −5.67421478804982519747588140235, −5.55937071153567886442405128528, −5.28236838001925575274718265820, −5.24253086285196935427044206751, −5.22572352398713312068158411587, −4.54476408260015526892293406301, −4.33047987072581656482729135933, −4.19509184994293579437310821136, −3.22803982532504872327437878837, −3.20554758924775429525750760784, −3.15476334098191105333761340239, −2.83998557278829277960114792903, −2.59763227495288490600146869405, −2.35140364477098046879374861839, −2.17469550128057165514208641760, −1.31752065089440526798832245761, −1.19824296260177998244584975543, −0.835750787407317372910257592154, −0.66756268367327806591977822754, −0.41144271372840403216473021353, −0.05902993498742956098510830027, 0.05902993498742956098510830027, 0.41144271372840403216473021353, 0.66756268367327806591977822754, 0.835750787407317372910257592154, 1.19824296260177998244584975543, 1.31752065089440526798832245761, 2.17469550128057165514208641760, 2.35140364477098046879374861839, 2.59763227495288490600146869405, 2.83998557278829277960114792903, 3.15476334098191105333761340239, 3.20554758924775429525750760784, 3.22803982532504872327437878837, 4.19509184994293579437310821136, 4.33047987072581656482729135933, 4.54476408260015526892293406301, 5.22572352398713312068158411587, 5.24253086285196935427044206751, 5.28236838001925575274718265820, 5.55937071153567886442405128528, 5.67421478804982519747588140235, 5.68848182162305454268147159134, 6.50632904210740071021581079855, 6.51067907453677788855868915565, 6.55563985544091262671822436038
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.