Dirichlet series
| L(s) = 1 | + 4·2-s + 9·4-s + 2·5-s − 7-s + 14·8-s + 6·9-s + 8·10-s + 11·11-s − 4·14-s + 18·16-s + 5·17-s + 24·18-s + 9·19-s + 18·20-s + 44·22-s − 10·23-s + 10·25-s − 9·28-s − 6·29-s − 6·31-s + 22·32-s + 20·34-s − 2·35-s + 54·36-s + 4·37-s + 36·38-s + 28·40-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 9/2·4-s + 0.894·5-s − 0.377·7-s + 4.94·8-s + 2·9-s + 2.52·10-s + 3.31·11-s − 1.06·14-s + 9/2·16-s + 1.21·17-s + 5.65·18-s + 2.06·19-s + 4.02·20-s + 9.38·22-s − 2.08·23-s + 2·25-s − 1.70·28-s − 1.11·29-s − 1.07·31-s + 3.88·32-s + 3.42·34-s − 0.338·35-s + 9·36-s + 0.657·37-s + 5.83·38-s + 4.42·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(7^{10} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.65742\times 10^{9}\) |
| Root analytic conductor: | \(3.07348\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 7^{10} \cdot 13^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(91.19061478\) |
| \(L(\frac12)\) | \(\approx\) | \(91.19061478\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 + T + 6 T^{2} + 17 T^{3} + 17 T^{4} + 204 T^{5} + 17 p T^{6} + 17 p^{2} T^{7} + 6 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p^{2} T + 7 T^{2} - 3 p T^{3} - T^{4} + 5 p T^{5} - 9 T^{6} - p^{3} T^{7} + 31 T^{8} - 31 p T^{9} + 101 T^{10} - 31 p^{2} T^{11} + 31 p^{2} T^{12} - p^{6} T^{13} - 9 p^{4} T^{14} + 5 p^{6} T^{15} - p^{6} T^{16} - 3 p^{8} T^{17} + 7 p^{8} T^{18} - p^{11} T^{19} + p^{10} T^{20} \) |
| 3 | \( 1 - 2 p T^{2} + 11 T^{4} - 4 T^{5} + 16 p T^{7} + 13 T^{8} - 100 T^{9} - 158 T^{10} - 100 p T^{11} + 13 p^{2} T^{12} + 16 p^{4} T^{13} - 4 p^{5} T^{15} + 11 p^{6} T^{16} - 2 p^{9} T^{18} + p^{10} T^{20} \) | |
| 5 | \( 1 - 2 T - 6 T^{2} + 4 p T^{3} - 13 T^{4} - 56 T^{5} + 104 T^{6} + 54 p T^{7} - 531 T^{8} - 1204 T^{9} + 4634 T^{10} - 1204 p T^{11} - 531 p^{2} T^{12} + 54 p^{4} T^{13} + 104 p^{4} T^{14} - 56 p^{5} T^{15} - 13 p^{6} T^{16} + 4 p^{8} T^{17} - 6 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - p T + 30 T^{2} + p T^{3} + 362 T^{4} - 247 p T^{5} + 872 T^{6} - 93 p T^{7} + 137985 T^{8} - 21860 p T^{9} - 53516 p T^{10} - 21860 p^{2} T^{11} + 137985 p^{2} T^{12} - 93 p^{4} T^{13} + 872 p^{4} T^{14} - 247 p^{6} T^{15} + 362 p^{6} T^{16} + p^{8} T^{17} + 30 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 5 T - 38 T^{2} + 9 p T^{3} + 938 T^{4} - 1627 T^{5} - 26688 T^{6} + 31085 T^{7} + 551101 T^{8} - 392902 T^{9} - 9014404 T^{10} - 392902 p T^{11} + 551101 p^{2} T^{12} + 31085 p^{3} T^{13} - 26688 p^{4} T^{14} - 1627 p^{5} T^{15} + 938 p^{6} T^{16} + 9 p^{8} T^{17} - 38 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 9 T + 287 T^{3} - 996 T^{4} - 447 T^{5} + 12448 T^{6} - 99505 T^{7} + 442983 T^{8} + 1222014 T^{9} - 15728840 T^{10} + 1222014 p T^{11} + 442983 p^{2} T^{12} - 99505 p^{3} T^{13} + 12448 p^{4} T^{14} - 447 p^{5} T^{15} - 996 p^{6} T^{16} + 287 p^{7} T^{17} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 10 T - 2 p T^{2} - 432 T^{3} + 193 p T^{4} + 22700 T^{5} - 176024 T^{6} - 390982 T^{7} + 7191797 T^{8} + 6669500 T^{9} - 170411134 T^{10} + 6669500 p T^{11} + 7191797 p^{2} T^{12} - 390982 p^{3} T^{13} - 176024 p^{4} T^{14} + 22700 p^{5} T^{15} + 193 p^{7} T^{16} - 432 p^{7} T^{17} - 2 p^{9} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( ( 1 + 3 T + 120 T^{2} + 329 T^{3} + 6379 T^{4} + 13928 T^{5} + 6379 p T^{6} + 329 p^{2} T^{7} + 120 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 31 | \( 1 + 6 T - 58 T^{2} - 720 T^{3} + 539 T^{4} + 34924 T^{5} + 102428 T^{6} - 950786 T^{7} - 6513587 T^{8} + 10872140 T^{9} + 235305322 T^{10} + 10872140 p T^{11} - 6513587 p^{2} T^{12} - 950786 p^{3} T^{13} + 102428 p^{4} T^{14} + 34924 p^{5} T^{15} + 539 p^{6} T^{16} - 720 p^{7} T^{17} - 58 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 4 T - 58 T^{2} - 468 T^{3} + 3791 T^{4} + 32876 T^{5} + 48516 T^{6} - 1529504 T^{7} - 7012835 T^{8} + 16657880 T^{9} + 398651282 T^{10} + 16657880 p T^{11} - 7012835 p^{2} T^{12} - 1529504 p^{3} T^{13} + 48516 p^{4} T^{14} + 32876 p^{5} T^{15} + 3791 p^{6} T^{16} - 468 p^{7} T^{17} - 58 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( ( 1 + 14 T + 177 T^{2} + 1356 T^{3} + 10822 T^{4} + 65708 T^{5} + 10822 p T^{6} + 1356 p^{2} T^{7} + 177 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 43 | \( ( 1 - 2 T + 143 T^{2} - 36 T^{3} + 8914 T^{4} + 4364 T^{5} + 8914 p T^{6} - 36 p^{2} T^{7} + 143 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( 1 - T - 110 T^{2} + 213 T^{3} + 4742 T^{4} - 19127 T^{5} - 115824 T^{6} + 1639327 T^{7} + 2920165 T^{8} - 47057396 T^{9} - 64021660 T^{10} - 47057396 p T^{11} + 2920165 p^{2} T^{12} + 1639327 p^{3} T^{13} - 115824 p^{4} T^{14} - 19127 p^{5} T^{15} + 4742 p^{6} T^{16} + 213 p^{7} T^{17} - 110 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 17 T + 98 T^{2} + 891 T^{3} + 12890 T^{4} + 113191 T^{5} + 924808 T^{6} + 7128691 T^{7} + 49614557 T^{8} + 435513922 T^{9} + 3680642300 T^{10} + 435513922 p T^{11} + 49614557 p^{2} T^{12} + 7128691 p^{3} T^{13} + 924808 p^{4} T^{14} + 113191 p^{5} T^{15} + 12890 p^{6} T^{16} + 891 p^{7} T^{17} + 98 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 11 T - 210 T^{2} + 1595 T^{3} + 39626 T^{4} - 193853 T^{5} - 4463656 T^{6} + 10280721 T^{7} + 413214561 T^{8} - 345980140 T^{9} - 26912019076 T^{10} - 345980140 p T^{11} + 413214561 p^{2} T^{12} + 10280721 p^{3} T^{13} - 4463656 p^{4} T^{14} - 193853 p^{5} T^{15} + 39626 p^{6} T^{16} + 1595 p^{7} T^{17} - 210 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 11 T - 62 T^{2} + 1823 T^{3} - 8374 T^{4} - 37837 T^{5} + 831656 T^{6} - 8313145 T^{7} + 39294509 T^{8} + 417204170 T^{9} - 6666109444 T^{10} + 417204170 p T^{11} + 39294509 p^{2} T^{12} - 8313145 p^{3} T^{13} + 831656 p^{4} T^{14} - 37837 p^{5} T^{15} - 8374 p^{6} T^{16} + 1823 p^{7} T^{17} - 62 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 13 T - 4 T^{2} + 399 T^{3} + 980 T^{4} - 7291 T^{5} + 211020 T^{6} - 2849405 T^{7} - 6211985 T^{8} - 77247214 T^{9} + 3104294312 T^{10} - 77247214 p T^{11} - 6211985 p^{2} T^{12} - 2849405 p^{3} T^{13} + 211020 p^{4} T^{14} - 7291 p^{5} T^{15} + 980 p^{6} T^{16} + 399 p^{7} T^{17} - 4 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( ( 1 + 15 T + 330 T^{2} + 3407 T^{3} + 44629 T^{4} + 338900 T^{5} + 44629 p T^{6} + 3407 p^{2} T^{7} + 330 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( 1 - 290 T^{2} - 84 T^{3} + 46535 T^{4} + 17600 T^{5} - 5407596 T^{6} - 1342052 T^{7} + 502835605 T^{8} + 38160940 T^{9} - 39632869574 T^{10} + 38160940 p T^{11} + 502835605 p^{2} T^{12} - 1342052 p^{3} T^{13} - 5407596 p^{4} T^{14} + 17600 p^{5} T^{15} + 46535 p^{6} T^{16} - 84 p^{7} T^{17} - 290 p^{8} T^{18} + p^{10} T^{20} \) | |
| 79 | \( 1 + 2 T - 254 T^{2} - 1128 T^{3} + 33479 T^{4} + 192224 T^{5} - 2680240 T^{6} - 18708310 T^{7} + 153962701 T^{8} + 668721992 T^{9} - 8689534830 T^{10} + 668721992 p T^{11} + 153962701 p^{2} T^{12} - 18708310 p^{3} T^{13} - 2680240 p^{4} T^{14} + 192224 p^{5} T^{15} + 33479 p^{6} T^{16} - 1128 p^{7} T^{17} - 254 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( ( 1 + 6 T + 291 T^{2} + 1684 T^{3} + 40702 T^{4} + 204364 T^{5} + 40702 p T^{6} + 1684 p^{2} T^{7} + 291 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 + 4 T - 274 T^{2} - 300 T^{3} + 463 p T^{4} - 380 p T^{5} - 4241132 T^{6} + 6119888 T^{7} + 353742509 T^{8} - 232834936 T^{9} - 28925933590 T^{10} - 232834936 p T^{11} + 353742509 p^{2} T^{12} + 6119888 p^{3} T^{13} - 4241132 p^{4} T^{14} - 380 p^{6} T^{15} + 463 p^{7} T^{16} - 300 p^{7} T^{17} - 274 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( ( 1 - 12 T + 469 T^{2} - 4044 T^{3} + 87194 T^{4} - 556336 T^{5} + 87194 p T^{6} - 4044 p^{2} T^{7} + 469 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
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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
−3.53017543353884194571818430951, −3.51974463214059652768854227097, −3.41244324716963943932487955176, −3.35349231667439640073397596676, −3.24685210453364790969797195146, −3.14082863159984650513535347444, −3.10648867390628805538442270541, −2.84468637507219491462951446611, −2.68913694026208346189590378780, −2.42198830365041545671222303839, −2.39366174369050923807680320537, −2.33080952523630176432825047521, −2.26397379009176753480227391705, −2.20831483594276703939584522927, −1.87341689559828041142864780259, −1.66794665270985384190261762004, −1.62457855102682437152259631607, −1.42683207716793274764940884928, −1.41936383797075055609014222901, −1.38739928150821839924834340663, −1.10190866919114607791222693522, −1.08085235607704099021025455737, −1.07337059955423013253447857567, −0.34413024237294033164936519996, −0.32216117794876037745528135762, 0.32216117794876037745528135762, 0.34413024237294033164936519996, 1.07337059955423013253447857567, 1.08085235607704099021025455737, 1.10190866919114607791222693522, 1.38739928150821839924834340663, 1.41936383797075055609014222901, 1.42683207716793274764940884928, 1.62457855102682437152259631607, 1.66794665270985384190261762004, 1.87341689559828041142864780259, 2.20831483594276703939584522927, 2.26397379009176753480227391705, 2.33080952523630176432825047521, 2.39366174369050923807680320537, 2.42198830365041545671222303839, 2.68913694026208346189590378780, 2.84468637507219491462951446611, 3.10648867390628805538442270541, 3.14082863159984650513535347444, 3.24685210453364790969797195146, 3.35349231667439640073397596676, 3.41244324716963943932487955176, 3.51974463214059652768854227097, 3.53017543353884194571818430951
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.