Dirichlet series
| L(s) = 1 | − 5.16e4·9-s + 2.77e4·11-s − 2.92e6·23-s − 8.02e6·25-s − 8.35e5·29-s − 3.19e7·37-s + 1.34e7·43-s + 1.37e8·53-s − 3.42e8·67-s − 5.31e8·71-s + 4.62e8·79-s + 1.49e9·81-s − 1.43e9·99-s + 8.67e9·107-s + 6.04e9·109-s − 5.45e9·113-s + 1.16e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.28e10·169-s + ⋯ |
| L(s) = 1 | − 2.62·9-s + 0.571·11-s − 2.17·23-s − 4.10·25-s − 0.219·29-s − 2.80·37-s + 0.598·43-s + 2.39·53-s − 2.07·67-s − 2.48·71-s + 1.33·79-s + 3.86·81-s − 1.49·99-s + 6.39·107-s + 4.10·109-s − 3.14·113-s + 0.0493·121-s − 4.04·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{20} \cdot 7^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.09884\times 10^{20}\) |
| Root analytic conductor: | \(10.0472\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{20} \cdot 7^{20} ,\ ( \ : [9/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(3.532712391\) |
| \(L(\frac12)\) | \(\approx\) | \(3.532712391\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + 51640 T^{2} + 129991913 p^{2} T^{4} + 89386357616 p^{5} T^{6} + 932659510272902 p^{6} T^{8} + 2649123243151909936 p^{8} T^{10} + 932659510272902 p^{24} T^{12} + 89386357616 p^{41} T^{14} + 129991913 p^{56} T^{16} + 51640 p^{72} T^{18} + p^{90} T^{20} \) |
| 5 | \( 1 + 8023126 T^{2} + 7557866575509 p T^{4} + \)\(12\!\cdots\!28\)\( T^{6} + \)\(13\!\cdots\!42\)\( p^{2} T^{8} + \)\(11\!\cdots\!12\)\( p^{4} T^{10} + \)\(13\!\cdots\!42\)\( p^{20} T^{12} + \)\(12\!\cdots\!28\)\( p^{36} T^{14} + 7557866575509 p^{55} T^{16} + 8023126 p^{72} T^{18} + p^{90} T^{20} \) | |
| 11 | \( ( 1 - 13870 T + 230380307 T^{2} - 121416084737424 T^{3} + 4071722023471596662 T^{4} + \)\(17\!\cdots\!20\)\( T^{5} + 4071722023471596662 p^{9} T^{6} - 121416084737424 p^{18} T^{7} + 230380307 p^{27} T^{8} - 13870 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 13 | \( 1 + 42872358142 T^{2} + \)\(10\!\cdots\!65\)\( T^{4} + \)\(19\!\cdots\!04\)\( T^{6} + \)\(26\!\cdots\!78\)\( T^{8} + \)\(31\!\cdots\!40\)\( T^{10} + \)\(26\!\cdots\!78\)\( p^{18} T^{12} + \)\(19\!\cdots\!04\)\( p^{36} T^{14} + \)\(10\!\cdots\!65\)\( p^{54} T^{16} + 42872358142 p^{72} T^{18} + p^{90} T^{20} \) | |
| 17 | \( 1 + 817322314240 T^{2} + \)\(30\!\cdots\!45\)\( T^{4} + \)\(69\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!90\)\( T^{8} + \)\(14\!\cdots\!72\)\( T^{10} + \)\(11\!\cdots\!90\)\( p^{18} T^{12} + \)\(69\!\cdots\!60\)\( p^{36} T^{14} + \)\(30\!\cdots\!45\)\( p^{54} T^{16} + 817322314240 p^{72} T^{18} + p^{90} T^{20} \) | |
| 19 | \( 1 + 1069096244424 T^{2} + \)\(53\!\cdots\!57\)\( T^{4} + \)\(11\!\cdots\!76\)\( T^{6} - \)\(13\!\cdots\!34\)\( T^{8} - \)\(13\!\cdots\!56\)\( T^{10} - \)\(13\!\cdots\!34\)\( p^{18} T^{12} + \)\(11\!\cdots\!76\)\( p^{36} T^{14} + \)\(53\!\cdots\!57\)\( p^{54} T^{16} + 1069096244424 p^{72} T^{18} + p^{90} T^{20} \) | |
| 23 | \( ( 1 + 1460548 T + 4933120661267 T^{2} + 7393259636555450352 T^{3} + \)\(13\!\cdots\!98\)\( T^{4} + \)\(18\!\cdots\!16\)\( T^{5} + \)\(13\!\cdots\!98\)\( p^{9} T^{6} + 7393259636555450352 p^{18} T^{7} + 4933120661267 p^{27} T^{8} + 1460548 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 29 | \( ( 1 + 417874 T + 25626991612349 T^{2} - 68455178832265577088 T^{3} + \)\(15\!\cdots\!70\)\( T^{4} - \)\(21\!\cdots\!80\)\( T^{5} + \)\(15\!\cdots\!70\)\( p^{9} T^{6} - 68455178832265577088 p^{18} T^{7} + 25626991612349 p^{27} T^{8} + 417874 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 31 | \( 1 + 148568757342814 T^{2} + \)\(10\!\cdots\!73\)\( T^{4} + \)\(13\!\cdots\!08\)\( p T^{6} + \)\(12\!\cdots\!34\)\( T^{8} + \)\(32\!\cdots\!32\)\( T^{10} + \)\(12\!\cdots\!34\)\( p^{18} T^{12} + \)\(13\!\cdots\!08\)\( p^{37} T^{14} + \)\(10\!\cdots\!73\)\( p^{54} T^{16} + 148568757342814 p^{72} T^{18} + p^{90} T^{20} \) | |
| 37 | \( ( 1 + 15964378 T + 522610033736325 T^{2} + \)\(47\!\cdots\!72\)\( T^{3} + \)\(99\!\cdots\!62\)\( T^{4} + \)\(66\!\cdots\!08\)\( T^{5} + \)\(99\!\cdots\!62\)\( p^{9} T^{6} + \)\(47\!\cdots\!72\)\( p^{18} T^{7} + 522610033736325 p^{27} T^{8} + 15964378 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 41 | \( 1 + 1291137340447408 T^{2} + \)\(10\!\cdots\!73\)\( T^{4} + \)\(58\!\cdots\!88\)\( T^{6} + \)\(26\!\cdots\!18\)\( T^{8} + \)\(94\!\cdots\!72\)\( T^{10} + \)\(26\!\cdots\!18\)\( p^{18} T^{12} + \)\(58\!\cdots\!88\)\( p^{36} T^{14} + \)\(10\!\cdots\!73\)\( p^{54} T^{16} + 1291137340447408 p^{72} T^{18} + p^{90} T^{20} \) | |
| 43 | \( ( 1 - 6707170 T + 427676884211763 T^{2} - \)\(48\!\cdots\!24\)\( T^{3} + \)\(38\!\cdots\!66\)\( T^{4} - \)\(56\!\cdots\!08\)\( T^{5} + \)\(38\!\cdots\!66\)\( p^{9} T^{6} - \)\(48\!\cdots\!24\)\( p^{18} T^{7} + 427676884211763 p^{27} T^{8} - 6707170 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 47 | \( 1 + 4639166355403966 T^{2} + \)\(11\!\cdots\!65\)\( T^{4} + \)\(18\!\cdots\!72\)\( T^{6} + \)\(24\!\cdots\!06\)\( T^{8} + \)\(28\!\cdots\!88\)\( T^{10} + \)\(24\!\cdots\!06\)\( p^{18} T^{12} + \)\(18\!\cdots\!72\)\( p^{36} T^{14} + \)\(11\!\cdots\!65\)\( p^{54} T^{16} + 4639166355403966 p^{72} T^{18} + p^{90} T^{20} \) | |
| 53 | \( ( 1 - 68650894 T + 10155818033104481 T^{2} - \)\(44\!\cdots\!28\)\( T^{3} + \)\(43\!\cdots\!54\)\( T^{4} - \)\(14\!\cdots\!16\)\( T^{5} + \)\(43\!\cdots\!54\)\( p^{9} T^{6} - \)\(44\!\cdots\!28\)\( p^{18} T^{7} + 10155818033104481 p^{27} T^{8} - 68650894 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 59 | \( 1 + 36616895238447208 T^{2} + \)\(74\!\cdots\!05\)\( T^{4} + \)\(11\!\cdots\!40\)\( T^{6} + \)\(13\!\cdots\!02\)\( T^{8} + \)\(13\!\cdots\!08\)\( T^{10} + \)\(13\!\cdots\!02\)\( p^{18} T^{12} + \)\(11\!\cdots\!40\)\( p^{36} T^{14} + \)\(74\!\cdots\!05\)\( p^{54} T^{16} + 36616895238447208 p^{72} T^{18} + p^{90} T^{20} \) | |
| 61 | \( 1 + 86247106297271526 T^{2} + \)\(35\!\cdots\!49\)\( T^{4} + \)\(94\!\cdots\!64\)\( T^{6} + \)\(17\!\cdots\!86\)\( T^{8} + \)\(23\!\cdots\!72\)\( T^{10} + \)\(17\!\cdots\!86\)\( p^{18} T^{12} + \)\(94\!\cdots\!64\)\( p^{36} T^{14} + \)\(35\!\cdots\!49\)\( p^{54} T^{16} + 86247106297271526 p^{72} T^{18} + p^{90} T^{20} \) | |
| 67 | \( ( 1 + 171131268 T + 94867197787400351 T^{2} + \)\(12\!\cdots\!04\)\( T^{3} + \)\(44\!\cdots\!66\)\( T^{4} + \)\(46\!\cdots\!52\)\( T^{5} + \)\(44\!\cdots\!66\)\( p^{9} T^{6} + \)\(12\!\cdots\!04\)\( p^{18} T^{7} + 94867197787400351 p^{27} T^{8} + 171131268 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 71 | \( ( 1 + 265518204 T + 141014013985826707 T^{2} + \)\(34\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!06\)\( T^{4} + \)\(20\!\cdots\!24\)\( T^{5} + \)\(11\!\cdots\!06\)\( p^{9} T^{6} + \)\(34\!\cdots\!76\)\( p^{18} T^{7} + 141014013985826707 p^{27} T^{8} + 265518204 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 73 | \( 1 + 440777533172972976 T^{2} + \)\(94\!\cdots\!65\)\( T^{4} + \)\(12\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!02\)\( T^{8} + \)\(82\!\cdots\!16\)\( T^{10} + \)\(12\!\cdots\!02\)\( p^{18} T^{12} + \)\(12\!\cdots\!20\)\( p^{36} T^{14} + \)\(94\!\cdots\!65\)\( p^{54} T^{16} + 440777533172972976 p^{72} T^{18} + p^{90} T^{20} \) | |
| 79 | \( ( 1 - 231093292 T + 430967983148468475 T^{2} - \)\(83\!\cdots\!84\)\( T^{3} + \)\(89\!\cdots\!98\)\( T^{4} - \)\(17\!\cdots\!00\)\( p T^{5} + \)\(89\!\cdots\!98\)\( p^{9} T^{6} - \)\(83\!\cdots\!84\)\( p^{18} T^{7} + 430967983148468475 p^{27} T^{8} - 231093292 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 83 | \( 1 + 1222722994923789112 T^{2} + \)\(73\!\cdots\!49\)\( T^{4} + \)\(28\!\cdots\!92\)\( T^{6} + \)\(80\!\cdots\!94\)\( T^{8} + \)\(17\!\cdots\!96\)\( T^{10} + \)\(80\!\cdots\!94\)\( p^{18} T^{12} + \)\(28\!\cdots\!92\)\( p^{36} T^{14} + \)\(73\!\cdots\!49\)\( p^{54} T^{16} + 1222722994923789112 p^{72} T^{18} + p^{90} T^{20} \) | |
| 89 | \( 1 + 2375249502133647424 T^{2} + \)\(27\!\cdots\!17\)\( T^{4} + \)\(20\!\cdots\!08\)\( T^{6} + \)\(10\!\cdots\!06\)\( T^{8} + \)\(44\!\cdots\!68\)\( T^{10} + \)\(10\!\cdots\!06\)\( p^{18} T^{12} + \)\(20\!\cdots\!08\)\( p^{36} T^{14} + \)\(27\!\cdots\!17\)\( p^{54} T^{16} + 2375249502133647424 p^{72} T^{18} + p^{90} T^{20} \) | |
| 97 | \( 1 + 1343606629094988672 T^{2} + \)\(19\!\cdots\!05\)\( T^{4} + \)\(20\!\cdots\!24\)\( T^{6} + \)\(20\!\cdots\!22\)\( T^{8} + \)\(15\!\cdots\!24\)\( T^{10} + \)\(20\!\cdots\!22\)\( p^{18} T^{12} + \)\(20\!\cdots\!24\)\( p^{36} T^{14} + \)\(19\!\cdots\!05\)\( p^{54} T^{16} + 1343606629094988672 p^{72} T^{18} + p^{90} T^{20} \) | |
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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
−3.24089112847323656475693962108, −3.22134164189592353700326200847, −3.16245272593570775411788721527, −2.99506739677341213110259059926, −2.87018152735283679080532229830, −2.41617635063648988687737590703, −2.33249819837686277027443351603, −2.29535359585596281676413135040, −2.09099872790193974776395998576, −1.99280471055444261368370554944, −1.94548173909683589025499632451, −1.93045728546061518236080733251, −1.89959241464360249036573779161, −1.62472846277711246678360323275, −1.56179754844755238936617836285, −1.44459389110197677557400860987, −1.08098618736659732045769738392, −0.829251027335053799159110502979, −0.70877819848339789188764106605, −0.59777790389718912441119534269, −0.59285420916694119591320472411, −0.57566573447367152162130032585, −0.30673848028177719521176253147, −0.26831902065750811676775977306, −0.10060119059536151058407553485, 0.10060119059536151058407553485, 0.26831902065750811676775977306, 0.30673848028177719521176253147, 0.57566573447367152162130032585, 0.59285420916694119591320472411, 0.59777790389718912441119534269, 0.70877819848339789188764106605, 0.829251027335053799159110502979, 1.08098618736659732045769738392, 1.44459389110197677557400860987, 1.56179754844755238936617836285, 1.62472846277711246678360323275, 1.89959241464360249036573779161, 1.93045728546061518236080733251, 1.94548173909683589025499632451, 1.99280471055444261368370554944, 2.09099872790193974776395998576, 2.29535359585596281676413135040, 2.33249819837686277027443351603, 2.41617635063648988687737590703, 2.87018152735283679080532229830, 2.99506739677341213110259059926, 3.16245272593570775411788721527, 3.22134164189592353700326200847, 3.24089112847323656475693962108
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.