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 + 10·13-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 − 40·26-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 + ⋯ |
| 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 + 2.77·13-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 − 7.84·26-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 + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(7^{10} \cdot 13^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0410378\) |
| Root analytic conductor: | \(0.852431\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 7^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1771737816\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1771737816\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T )^{10} \) | |
| 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
−5.91328389419399910801102106711, −5.80794956159442223979761458822, −5.59642784637144022373725924927, −5.40016663764971863379182553645, −5.34784786711109292399120711796, −5.31618080681637123914031186301, −4.76004094709702901045631407632, −4.74040104744342984958208378829, −4.73310608718834228244856945712, −4.37162808524744484568853269875, −4.24429525564441193119084043889, −4.18767110669254397647783736591, −3.91118410282086055755930759134, −3.87695317706830688773733718277, −3.51892770271674247484985365349, −3.31649817649868200497682542822, −3.15985279368455953602078928041, −2.92050571152363376177525191301, −2.75840817566795589164627141121, −2.36060517314558161997451118530, −2.14399081386139421756398149636, −1.89202839583541384462979097762, −1.82135772864216292595207452466, −1.12783726141663635359438766510, −0.993478228471500774835515713466, 0.993478228471500774835515713466, 1.12783726141663635359438766510, 1.82135772864216292595207452466, 1.89202839583541384462979097762, 2.14399081386139421756398149636, 2.36060517314558161997451118530, 2.75840817566795589164627141121, 2.92050571152363376177525191301, 3.15985279368455953602078928041, 3.31649817649868200497682542822, 3.51892770271674247484985365349, 3.87695317706830688773733718277, 3.91118410282086055755930759134, 4.18767110669254397647783736591, 4.24429525564441193119084043889, 4.37162808524744484568853269875, 4.73310608718834228244856945712, 4.74040104744342984958208378829, 4.76004094709702901045631407632, 5.31618080681637123914031186301, 5.34784786711109292399120711796, 5.40016663764971863379182553645, 5.59642784637144022373725924927, 5.80794956159442223979761458822, 5.91328389419399910801102106711
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.