Dirichlet series
| L(s) = 1 | − 7·2-s − 7·3-s + 24·4-s − 3·5-s + 49·6-s − 5·7-s − 55·8-s + 25·9-s + 21·10-s + 7·11-s − 168·12-s − 3·13-s + 35·14-s + 21·15-s + 99·16-s − 10·17-s − 175·18-s + 2·19-s − 72·20-s + 35·21-s − 49·22-s − 12·23-s + 385·24-s + 5·25-s + 21·26-s − 66·27-s − 120·28-s + ⋯ |
| L(s) = 1 | − 4.94·2-s − 4.04·3-s + 12·4-s − 1.34·5-s + 20.0·6-s − 1.88·7-s − 19.4·8-s + 25/3·9-s + 6.64·10-s + 2.11·11-s − 48.4·12-s − 0.832·13-s + 9.35·14-s + 5.42·15-s + 99/4·16-s − 2.42·17-s − 41.2·18-s + 0.458·19-s − 16.0·20-s + 7.63·21-s − 10.4·22-s − 2.50·23-s + 78.5·24-s + 25-s + 4.11·26-s − 12.7·27-s − 22.6·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.36565\times 10^{-8}\) |
| Root analytic conductor: | \(0.428550\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.0005213496287\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0005213496287\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 23 | \( 1 + 12 T - 10 T^{2} - 527 T^{3} - 32 T^{4} + 14103 T^{5} - 32 p T^{6} - 527 p^{2} T^{7} - 10 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) |
| good | 2 | \( 1 + 7 T + 25 T^{2} + 31 p T^{3} + 15 p^{3} T^{4} + 47 p^{2} T^{5} + 229 T^{6} + 91 p T^{7} + p T^{8} - 71 p^{2} T^{9} - 551 T^{10} - 71 p^{3} T^{11} + p^{3} T^{12} + 91 p^{4} T^{13} + 229 p^{4} T^{14} + 47 p^{7} T^{15} + 15 p^{9} T^{16} + 31 p^{8} T^{17} + 25 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) |
| 3 | \( 1 + 7 T + 8 p T^{2} + 59 T^{3} + 143 T^{4} + 362 T^{5} + 829 T^{6} + 1637 T^{7} + 3076 T^{8} + 1969 p T^{9} + 10825 T^{10} + 1969 p^{2} T^{11} + 3076 p^{2} T^{12} + 1637 p^{3} T^{13} + 829 p^{4} T^{14} + 362 p^{5} T^{15} + 143 p^{6} T^{16} + 59 p^{7} T^{17} + 8 p^{9} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 5 | \( 1 + 3 T + 4 T^{2} + 19 T^{3} + 37 T^{4} + 38 T^{5} + 259 T^{6} + 587 T^{7} + 1214 T^{8} + 3853 T^{9} + 9119 T^{10} + 3853 p T^{11} + 1214 p^{2} T^{12} + 587 p^{3} T^{13} + 259 p^{4} T^{14} + 38 p^{5} T^{15} + 37 p^{6} T^{16} + 19 p^{7} T^{17} + 4 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 + 5 T + 18 T^{2} + 88 T^{3} + 347 T^{4} + 1075 T^{5} + 3628 T^{6} + 1650 p T^{7} + 31166 T^{8} + 88994 T^{9} + 252361 T^{10} + 88994 p T^{11} + 31166 p^{2} T^{12} + 1650 p^{4} T^{13} + 3628 p^{4} T^{14} + 1075 p^{5} T^{15} + 347 p^{6} T^{16} + 88 p^{7} T^{17} + 18 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 7 T + 5 T^{2} + 64 T^{3} - 239 T^{4} + 243 T^{5} + 4448 T^{6} - 26384 T^{7} + 38597 T^{8} + 113996 T^{9} - 669129 T^{10} + 113996 p T^{11} + 38597 p^{2} T^{12} - 26384 p^{3} T^{13} + 4448 p^{4} T^{14} + 243 p^{5} T^{15} - 239 p^{6} T^{16} + 64 p^{7} T^{17} + 5 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 3 T - 4 T^{2} - 150 T^{3} - 57 T^{4} + 1141 T^{5} + 7530 T^{6} - 26970 T^{7} - 69482 T^{8} - 33220 T^{9} + 2163305 T^{10} - 33220 p T^{11} - 69482 p^{2} T^{12} - 26970 p^{3} T^{13} + 7530 p^{4} T^{14} + 1141 p^{5} T^{15} - 57 p^{6} T^{16} - 150 p^{7} T^{17} - 4 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 10 T + 39 T^{2} + 99 T^{3} - 36 T^{4} - 2241 T^{5} - 674 p T^{6} - 44847 T^{7} - 59633 T^{8} + 522161 T^{9} + 3253139 T^{10} + 522161 p T^{11} - 59633 p^{2} T^{12} - 44847 p^{3} T^{13} - 674 p^{5} T^{14} - 2241 p^{5} T^{15} - 36 p^{6} T^{16} + 99 p^{7} T^{17} + 39 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 2 T - 59 T^{2} + 310 T^{3} + 1513 T^{4} - 14460 T^{5} + 481 T^{6} + 343914 T^{7} - 1093143 T^{8} - 3113594 T^{9} + 31471397 T^{10} - 3113594 p T^{11} - 1093143 p^{2} T^{12} + 343914 p^{3} T^{13} + 481 p^{4} T^{14} - 14460 p^{5} T^{15} + 1513 p^{6} T^{16} + 310 p^{7} T^{17} - 59 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 14 T + 57 T^{2} - 458 T^{3} + 5001 T^{4} - 13986 T^{5} + 85645 T^{6} - 1071714 T^{7} + 1968673 T^{8} - 8608842 T^{9} + 157266187 T^{10} - 8608842 p T^{11} + 1968673 p^{2} T^{12} - 1071714 p^{3} T^{13} + 85645 p^{4} T^{14} - 13986 p^{5} T^{15} + 5001 p^{6} T^{16} - 458 p^{7} T^{17} + 57 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 10 T + 25 T^{2} - 226 T^{3} + 1925 T^{4} + 4982 T^{5} - 42879 T^{6} - 47226 T^{7} - 1407939 T^{8} + 6585942 T^{9} + 24254759 T^{10} + 6585942 p T^{11} - 1407939 p^{2} T^{12} - 47226 p^{3} T^{13} - 42879 p^{4} T^{14} + 4982 p^{5} T^{15} + 1925 p^{6} T^{16} - 226 p^{7} T^{17} + 25 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 19 T + 137 T^{2} + 118 T^{3} - 187 p T^{4} - 76207 T^{5} - 426550 T^{6} - 880182 T^{7} + 7739069 T^{8} + 96108188 T^{9} + 657904633 T^{10} + 96108188 p T^{11} + 7739069 p^{2} T^{12} - 880182 p^{3} T^{13} - 426550 p^{4} T^{14} - 76207 p^{5} T^{15} - 187 p^{7} T^{16} + 118 p^{7} T^{17} + 137 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 7 T - 25 T^{2} + 583 T^{3} - 2660 T^{4} + 2131 T^{5} + 107915 T^{6} - 669350 T^{7} + 206320 T^{8} + 11256041 T^{9} + 10368863 T^{10} + 11256041 p T^{11} + 206320 p^{2} T^{12} - 669350 p^{3} T^{13} + 107915 p^{4} T^{14} + 2131 p^{5} T^{15} - 2660 p^{6} T^{16} + 583 p^{7} T^{17} - 25 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 11 T + 23 T^{2} - 605 T^{3} - 5312 T^{4} - 29546 T^{5} - 40138 T^{6} + 747769 T^{7} + 9344490 T^{8} + 16662481 T^{9} - 37880567 T^{10} + 16662481 p T^{11} + 9344490 p^{2} T^{12} + 747769 p^{3} T^{13} - 40138 p^{4} T^{14} - 29546 p^{5} T^{15} - 5312 p^{6} T^{16} - 605 p^{7} T^{17} + 23 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 + 9 T + 230 T^{2} + 1595 T^{3} + 21279 T^{4} + 110169 T^{5} + 21279 p T^{6} + 1595 p^{2} T^{7} + 230 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 - 29 T + 425 T^{2} - 3671 T^{3} + 13160 T^{4} + 2200 p T^{5} - 2282954 T^{6} + 17208509 T^{7} - 42065902 T^{8} - 503004225 T^{9} + 6387620997 T^{10} - 503004225 p T^{11} - 42065902 p^{2} T^{12} + 17208509 p^{3} T^{13} - 2282954 p^{4} T^{14} + 2200 p^{6} T^{15} + 13160 p^{6} T^{16} - 3671 p^{7} T^{17} + 425 p^{8} T^{18} - 29 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 21 T + 206 T^{2} + 2416 T^{3} + 25558 T^{4} + 134475 T^{5} + 585400 T^{6} + 1747337 T^{7} - 63466585 T^{8} - 768235069 T^{9} - 4979757289 T^{10} - 768235069 p T^{11} - 63466585 p^{2} T^{12} + 1747337 p^{3} T^{13} + 585400 p^{4} T^{14} + 134475 p^{5} T^{15} + 25558 p^{6} T^{16} + 2416 p^{7} T^{17} + 206 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 3 T + 157 T^{2} - 695 T^{3} + 17192 T^{4} - 102450 T^{5} + 1128572 T^{6} - 10677101 T^{7} + 64622634 T^{8} - 839211059 T^{9} + 3256596793 T^{10} - 839211059 p T^{11} + 64622634 p^{2} T^{12} - 10677101 p^{3} T^{13} + 1128572 p^{4} T^{14} - 102450 p^{5} T^{15} + 17192 p^{6} T^{16} - 695 p^{7} T^{17} + 157 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 45 T + 1023 T^{2} - 15905 T^{3} + 191575 T^{4} - 1861057 T^{5} + 14558842 T^{6} - 88766657 T^{7} + 371829810 T^{8} - 453912238 T^{9} - 4886911745 T^{10} - 453912238 p T^{11} + 371829810 p^{2} T^{12} - 88766657 p^{3} T^{13} + 14558842 p^{4} T^{14} - 1861057 p^{5} T^{15} + 191575 p^{6} T^{16} - 15905 p^{7} T^{17} + 1023 p^{8} T^{18} - 45 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 14 T + 59 T^{2} + 140 T^{3} + 3909 T^{4} + 54686 T^{5} - 152597 T^{6} - 4301040 T^{7} - 22667179 T^{8} - 212067842 T^{9} - 1811694401 T^{10} - 212067842 p T^{11} - 22667179 p^{2} T^{12} - 4301040 p^{3} T^{13} - 152597 p^{4} T^{14} + 54686 p^{5} T^{15} + 3909 p^{6} T^{16} + 140 p^{7} T^{17} + 59 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 19 T + 178 T^{2} - 444 T^{3} + 3164 T^{4} - 41113 T^{5} + 460294 T^{6} + 2252883 T^{7} - 34972209 T^{8} + 676028375 T^{9} - 5421480987 T^{10} + 676028375 p T^{11} - 34972209 p^{2} T^{12} + 2252883 p^{3} T^{13} + 460294 p^{4} T^{14} - 41113 p^{5} T^{15} + 3164 p^{6} T^{16} - 444 p^{7} T^{17} + 178 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 15 T + 36 T^{2} - 2262 T^{3} - 24256 T^{4} - 43814 T^{5} + 2049210 T^{6} + 237520 p T^{7} + 60623103 T^{8} - 1063681247 T^{9} - 12642166076 T^{10} - 1063681247 p T^{11} + 60623103 p^{2} T^{12} + 237520 p^{4} T^{13} + 2049210 p^{4} T^{14} - 43814 p^{5} T^{15} - 24256 p^{6} T^{16} - 2262 p^{7} T^{17} + 36 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 18 T + 109 T^{2} - 72 T^{3} - 10248 T^{4} + 168066 T^{5} - 1355830 T^{6} + 6119988 T^{7} + 37604796 T^{8} - 775038374 T^{9} + 6144118025 T^{10} - 775038374 p T^{11} + 37604796 p^{2} T^{12} + 6119988 p^{3} T^{13} - 1355830 p^{4} T^{14} + 168066 p^{5} T^{15} - 10248 p^{6} T^{16} - 72 p^{7} T^{17} + 109 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 25 T + 173 T^{2} + 1772 T^{3} - 48367 T^{4} + 349557 T^{5} + 2352100 T^{6} - 63458766 T^{7} + 346368857 T^{8} + 3220239858 T^{9} - 63184785399 T^{10} + 3220239858 p T^{11} + 346368857 p^{2} T^{12} - 63458766 p^{3} T^{13} + 2352100 p^{4} T^{14} + 349557 p^{5} T^{15} - 48367 p^{6} T^{16} + 1772 p^{7} T^{17} + 173 p^{8} T^{18} - 25 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 34 T + 333 T^{2} - 2173 T^{3} - 51172 T^{4} + 167353 T^{5} + 6267260 T^{6} - 25666611 T^{7} - 886006665 T^{8} + 2096067809 T^{9} + 115155056919 T^{10} + 2096067809 p T^{11} - 886006665 p^{2} T^{12} - 25666611 p^{3} T^{13} + 6267260 p^{4} T^{14} + 167353 p^{5} T^{15} - 51172 p^{6} T^{16} - 2173 p^{7} T^{17} + 333 p^{8} T^{18} + 34 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−8.343877245770678567496788060012, −8.245259285711677999907873202935, −8.230862233098849007811223164072, −7.72554860710428263572838304629, −7.62411840852442737605105286610, −7.27870100899071563448498861271, −7.21201196922533576535937855384, −6.84418738425911157201013021547, −6.72834303194291122857700768805, −6.71610085194056859926149333211, −6.51776563858356326443205144895, −6.49113510064450708961323541226, −6.43377255067923003768276281752, −6.09176650110013088563725642668, −5.79843252135569723671624138854, −5.46982253498209470819586318730, −5.43877314781797226185023792376, −5.02391178006753994249697831566, −4.86603750394554485763107657946, −4.28468710727745230810281531166, −4.27602166279250101438312391251, −4.02744855059760795335469833124, −3.53228341736896043965276327960, −3.17834018271689916685345129133, −2.19372870353046740349042896222, 2.19372870353046740349042896222, 3.17834018271689916685345129133, 3.53228341736896043965276327960, 4.02744855059760795335469833124, 4.27602166279250101438312391251, 4.28468710727745230810281531166, 4.86603750394554485763107657946, 5.02391178006753994249697831566, 5.43877314781797226185023792376, 5.46982253498209470819586318730, 5.79843252135569723671624138854, 6.09176650110013088563725642668, 6.43377255067923003768276281752, 6.49113510064450708961323541226, 6.51776563858356326443205144895, 6.71610085194056859926149333211, 6.72834303194291122857700768805, 6.84418738425911157201013021547, 7.21201196922533576535937855384, 7.27870100899071563448498861271, 7.62411840852442737605105286610, 7.72554860710428263572838304629, 8.230862233098849007811223164072, 8.245259285711677999907873202935, 8.343877245770678567496788060012
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.