Dirichlet series
| L(s) = 1 | + 2·3-s − 51·9-s − 2·13-s − 44·23-s − 25·25-s − 124·27-s − 46·29-s − 16·31-s − 4·39-s − 84·41-s − 112·47-s + 270·49-s + 262·59-s − 88·69-s − 236·71-s + 168·73-s − 50·75-s + 1.23e3·81-s − 92·87-s − 32·93-s − 198·101-s + 102·117-s + 635·121-s − 168·123-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2/3·3-s − 5.66·9-s − 0.153·13-s − 1.91·23-s − 25-s − 4.59·27-s − 1.58·29-s − 0.516·31-s − 0.102·39-s − 2.04·41-s − 2.38·47-s + 5.51·49-s + 4.44·59-s − 1.27·69-s − 3.32·71-s + 2.30·73-s − 2/3·75-s + 15.2·81-s − 1.05·87-s − 0.344·93-s − 1.96·101-s + 0.871·117-s + 5.24·121-s − 1.36·123-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{40} \cdot 5^{10} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.00352\times 10^{17}\) |
| Root analytic conductor: | \(7.08070\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{40} \cdot 5^{10} \cdot 23^{10} ,\ ( \ : [1]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(7.185936957\) |
| \(L(\frac12)\) | \(\approx\) | \(7.185936957\) |
| \(L(2)\) | 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 | \( ( 1 + p T^{2} )^{5} \) | |
| 23 | \( 1 + 44 T + 1153 T^{2} + 17856 T^{3} - 82 p^{2} T^{4} - 11864 p^{2} T^{5} - 82 p^{4} T^{6} + 17856 p^{4} T^{7} + 1153 p^{6} T^{8} + 44 p^{8} T^{9} + p^{10} T^{10} \) | |
| good | 3 | \( ( 1 - T + p^{3} T^{2} - 17 T^{3} + 41 p^{2} T^{4} - 146 T^{5} + 41 p^{4} T^{6} - 17 p^{4} T^{7} + p^{9} T^{8} - p^{8} T^{9} + p^{10} T^{10} )^{2} \) |
| 7 | \( 1 - 270 T^{2} + 37405 T^{4} - 3527995 T^{6} + 249083530 T^{8} - 13721583598 T^{10} + 249083530 p^{4} T^{12} - 3527995 p^{8} T^{14} + 37405 p^{12} T^{16} - 270 p^{16} T^{18} + p^{20} T^{20} \) | |
| 11 | \( 1 - 635 T^{2} + 217070 T^{4} - 50112170 T^{6} + 8679761185 T^{8} - 1176806007702 T^{10} + 8679761185 p^{4} T^{12} - 50112170 p^{8} T^{14} + 217070 p^{12} T^{16} - 635 p^{16} T^{18} + p^{20} T^{20} \) | |
| 13 | \( ( 1 + T + 697 T^{2} - 21 p T^{3} + 209489 T^{4} - 147124 T^{5} + 209489 p^{2} T^{6} - 21 p^{5} T^{7} + 697 p^{6} T^{8} + p^{8} T^{9} + p^{10} T^{10} )^{2} \) | |
| 17 | \( 1 - 570 T^{2} + 98605 T^{4} + 4389005 T^{6} - 2698345370 T^{8} + 18781200002 T^{10} - 2698345370 p^{4} T^{12} + 4389005 p^{8} T^{14} + 98605 p^{12} T^{16} - 570 p^{16} T^{18} + p^{20} T^{20} \) | |
| 19 | \( 1 - 1755 T^{2} + 1741630 T^{4} - 1206005530 T^{6} + 630409978705 T^{8} - 256188291168502 T^{10} + 630409978705 p^{4} T^{12} - 1206005530 p^{8} T^{14} + 1741630 p^{12} T^{16} - 1755 p^{16} T^{18} + p^{20} T^{20} \) | |
| 29 | \( ( 1 + 23 T + 3278 T^{2} + 71937 T^{3} + 4828603 T^{4} + 88945152 T^{5} + 4828603 p^{2} T^{6} + 71937 p^{4} T^{7} + 3278 p^{6} T^{8} + 23 p^{8} T^{9} + p^{10} T^{10} )^{2} \) | |
| 31 | \( ( 1 + 8 T + 3278 T^{2} + 31307 T^{3} + 5419183 T^{4} + 40810047 T^{5} + 5419183 p^{2} T^{6} + 31307 p^{4} T^{7} + 3278 p^{6} T^{8} + 8 p^{8} T^{9} + p^{10} T^{10} )^{2} \) | |
| 37 | \( 1 - 845 T^{2} + 5074205 T^{4} - 8656193420 T^{6} + 10882720856530 T^{8} - 26437808727606798 T^{10} + 10882720856530 p^{4} T^{12} - 8656193420 p^{8} T^{14} + 5074205 p^{12} T^{16} - 845 p^{16} T^{18} + p^{20} T^{20} \) | |
| 41 | \( ( 1 + 42 T + 3468 T^{2} + 33003 T^{3} + 120843 p T^{4} + 10975153 T^{5} + 120843 p^{3} T^{6} + 33003 p^{4} T^{7} + 3468 p^{6} T^{8} + 42 p^{8} T^{9} + p^{10} T^{10} )^{2} \) | |
| 43 | \( 1 - 9150 T^{2} + 45730765 T^{4} - 156922108360 T^{6} + 411434682849250 T^{8} - 850500408903448948 T^{10} + 411434682849250 p^{4} T^{12} - 156922108360 p^{8} T^{14} + 45730765 p^{12} T^{16} - 9150 p^{16} T^{18} + p^{20} T^{20} \) | |
| 47 | \( ( 1 + 56 T + 11632 T^{2} + 491952 T^{3} + 52652579 T^{4} + 1626995176 T^{5} + 52652579 p^{2} T^{6} + 491952 p^{4} T^{7} + 11632 p^{6} T^{8} + 56 p^{8} T^{9} + p^{10} T^{10} )^{2} \) | |
| 53 | \( 1 - 10165 T^{2} + 53761245 T^{4} - 230927530380 T^{6} + 858443049686610 T^{8} - 2645126205947297598 T^{10} + 858443049686610 p^{4} T^{12} - 230927530380 p^{8} T^{14} + 53761245 p^{12} T^{16} - 10165 p^{16} T^{18} + p^{20} T^{20} \) | |
| 59 | \( ( 1 - 131 T + 21185 T^{2} - 1788828 T^{3} + 160581238 T^{4} - 9279619170 T^{5} + 160581238 p^{2} T^{6} - 1788828 p^{4} T^{7} + 21185 p^{6} T^{8} - 131 p^{8} T^{9} + p^{10} T^{10} )^{2} \) | |
| 61 | \( 1 - 22235 T^{2} + 251666870 T^{4} - 1893757342970 T^{6} + 10465483940093785 T^{8} - 44256274238328891702 T^{10} + 10465483940093785 p^{4} T^{12} - 1893757342970 p^{8} T^{14} + 251666870 p^{12} T^{16} - 22235 p^{16} T^{18} + p^{20} T^{20} \) | |
| 67 | \( 1 - 25505 T^{2} + 334420625 T^{4} - 2955285070100 T^{6} + 19413750372449470 T^{8} - 98612788025953914198 T^{10} + 19413750372449470 p^{4} T^{12} - 2955285070100 p^{8} T^{14} + 334420625 p^{12} T^{16} - 25505 p^{16} T^{18} + p^{20} T^{20} \) | |
| 71 | \( ( 1 + 118 T + 14628 T^{2} + 1021917 T^{3} + 101491203 T^{4} + 94209537 p T^{5} + 101491203 p^{2} T^{6} + 1021917 p^{4} T^{7} + 14628 p^{6} T^{8} + 118 p^{8} T^{9} + p^{10} T^{10} )^{2} \) | |
| 73 | \( ( 1 - 84 T + 23952 T^{2} - 1315548 T^{3} + 224525619 T^{4} - 9069076424 T^{5} + 224525619 p^{2} T^{6} - 1315548 p^{4} T^{7} + 23952 p^{6} T^{8} - 84 p^{8} T^{9} + p^{10} T^{10} )^{2} \) | |
| 79 | \( 1 - 29790 T^{2} + 495654685 T^{4} - 5813141154760 T^{6} + 51785593788118690 T^{8} - \)\(36\!\cdots\!52\)\( T^{10} + 51785593788118690 p^{4} T^{12} - 5813141154760 p^{8} T^{14} + 495654685 p^{12} T^{16} - 29790 p^{16} T^{18} + p^{20} T^{20} \) | |
| 83 | \( 1 - 34765 T^{2} + 603018845 T^{4} - 7035545158380 T^{6} + 63282114771282610 T^{8} - \)\(47\!\cdots\!98\)\( T^{10} + 63282114771282610 p^{4} T^{12} - 7035545158380 p^{8} T^{14} + 603018845 p^{12} T^{16} - 34765 p^{16} T^{18} + p^{20} T^{20} \) | |
| 89 | \( 1 - 2990 T^{2} - 28174915 T^{4} - 450673037960 T^{6} + 3835140404307490 T^{8} - 1637663945943356052 T^{10} + 3835140404307490 p^{4} T^{12} - 450673037960 p^{8} T^{14} - 28174915 p^{12} T^{16} - 2990 p^{16} T^{18} + p^{20} T^{20} \) | |
| 97 | \( 1 - 33855 T^{2} + 683937550 T^{4} - 10378067123050 T^{6} + 128240370026012545 T^{8} - \)\(13\!\cdots\!98\)\( T^{10} + 128240370026012545 p^{4} T^{12} - 10378067123050 p^{8} T^{14} + 683937550 p^{12} T^{16} - 33855 p^{16} T^{18} + p^{20} T^{20} \) | |
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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
−3.11718762760679874638664405897, −2.89933403315843065108258651458, −2.88477146226456992039879542154, −2.86492417022157264886995808854, −2.59180227682824301438941407063, −2.55113509426416046864608780393, −2.45193631869849255763793599728, −2.35673582549591485522142334653, −2.34129306402285698157288182218, −2.01041619858425681819887553660, −1.99110832278896769632426364029, −1.97487783148514026825273432879, −1.89766202310115606380403235738, −1.66622224330387636413936419062, −1.66618874635500227322892903117, −1.57271643080993494055354333695, −1.41089095484184255939031043728, −0.931880792751963083851731295166, −0.70162178706136146337683827066, −0.67464478834258968312360569084, −0.60241406234677434875256181431, −0.46215906559862600960055601939, −0.41254608947918706970312038167, −0.39691555354390841236068968955, −0.17119675712263131078427052262, 0.17119675712263131078427052262, 0.39691555354390841236068968955, 0.41254608947918706970312038167, 0.46215906559862600960055601939, 0.60241406234677434875256181431, 0.67464478834258968312360569084, 0.70162178706136146337683827066, 0.931880792751963083851731295166, 1.41089095484184255939031043728, 1.57271643080993494055354333695, 1.66618874635500227322892903117, 1.66622224330387636413936419062, 1.89766202310115606380403235738, 1.97487783148514026825273432879, 1.99110832278896769632426364029, 2.01041619858425681819887553660, 2.34129306402285698157288182218, 2.35673582549591485522142334653, 2.45193631869849255763793599728, 2.55113509426416046864608780393, 2.59180227682824301438941407063, 2.86492417022157264886995808854, 2.88477146226456992039879542154, 2.89933403315843065108258651458, 3.11718762760679874638664405897
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.