Dirichlet series
| L(s) = 1 | − 4·2-s + 3-s + 5-s − 4·6-s − 3·7-s + 22·8-s + 11·9-s − 4·10-s + 4·11-s − 2·13-s + 12·14-s + 15-s − 32·16-s − 10·17-s − 44·18-s − 19-s − 3·21-s − 16·22-s + 2·23-s + 22·24-s + 19·25-s + 8·26-s + 10·27-s + 3·29-s − 4·30-s + 16·31-s − 20·32-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.577·3-s + 0.447·5-s − 1.63·6-s − 1.13·7-s + 7.77·8-s + 11/3·9-s − 1.26·10-s + 1.20·11-s − 0.554·13-s + 3.20·14-s + 0.258·15-s − 8·16-s − 2.42·17-s − 10.3·18-s − 0.229·19-s − 0.654·21-s − 3.41·22-s + 0.417·23-s + 4.49·24-s + 19/5·25-s + 1.56·26-s + 1.92·27-s + 0.557·29-s − 0.730·30-s + 2.87·31-s − 3.53·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0216681\) |
| Root analytic conductor: | \(0.852431\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.06716536305\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06716536305\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 + 3 T + 6 T^{2} + 45 T^{3} + 3 p^{2} T^{4} + 348 T^{5} + 1069 T^{6} + 348 p T^{7} + 3 p^{4} T^{8} + 45 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 2 T - 16 T^{2} + 3 T^{3} + 607 T^{4} + 433 T^{5} - 5615 T^{6} + 433 p T^{7} + 607 p^{2} T^{8} + 3 p^{3} T^{9} - 16 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( ( 1 + p T + 3 p T^{2} + 9 T^{3} + 5 p^{2} T^{4} + 7 p^{2} T^{5} + 51 T^{6} + 7 p^{3} T^{7} + 5 p^{4} T^{8} + 9 p^{3} T^{9} + 3 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2} \) |
| 3 | \( 1 - T - 10 T^{2} + 11 T^{3} + 53 T^{4} - 62 T^{5} - 167 T^{6} + 221 T^{7} + 98 p T^{8} - 535 T^{9} + 79 T^{10} + 604 T^{11} - 1559 T^{12} + 604 p T^{13} + 79 p^{2} T^{14} - 535 p^{3} T^{15} + 98 p^{5} T^{16} + 221 p^{5} T^{17} - 167 p^{6} T^{18} - 62 p^{7} T^{19} + 53 p^{8} T^{20} + 11 p^{9} T^{21} - 10 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 5 | \( 1 - T - 18 T^{2} - p T^{3} + 193 T^{4} + 192 T^{5} - 1181 T^{6} - 2139 T^{7} + 908 p T^{8} + 451 p^{2} T^{9} - 6679 T^{10} - 26266 T^{11} + 249 T^{12} - 26266 p T^{13} - 6679 p^{2} T^{14} + 451 p^{5} T^{15} + 908 p^{5} T^{16} - 2139 p^{5} T^{17} - 1181 p^{6} T^{18} + 192 p^{7} T^{19} + 193 p^{8} T^{20} - p^{10} T^{21} - 18 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 4 T - 29 T^{2} + 108 T^{3} + 477 T^{4} - 113 p T^{5} - 6686 T^{6} + 7665 T^{7} + 89323 T^{8} + 423 T^{9} - 1282040 T^{10} - 249219 T^{11} + 16505087 T^{12} - 249219 p T^{13} - 1282040 p^{2} T^{14} + 423 p^{3} T^{15} + 89323 p^{4} T^{16} + 7665 p^{5} T^{17} - 6686 p^{6} T^{18} - 113 p^{8} T^{19} + 477 p^{8} T^{20} + 108 p^{9} T^{21} - 29 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + 5 T + 90 T^{2} + 411 T^{3} + 3539 T^{4} + 13744 T^{5} + 78123 T^{6} + 13744 p T^{7} + 3539 p^{2} T^{8} + 411 p^{3} T^{9} + 90 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 + T - 49 T^{2} + 82 T^{3} + 1336 T^{4} - 4335 T^{5} - 10907 T^{6} + 99626 T^{7} - 263580 T^{8} - 1110690 T^{9} + 9684539 T^{10} + 2414194 T^{11} - 215227743 T^{12} + 2414194 p T^{13} + 9684539 p^{2} T^{14} - 1110690 p^{3} T^{15} - 263580 p^{4} T^{16} + 99626 p^{5} T^{17} - 10907 p^{6} T^{18} - 4335 p^{7} T^{19} + 1336 p^{8} T^{20} + 82 p^{9} T^{21} - 49 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - T + 32 T^{2} - 52 T^{3} + 1214 T^{4} - 1383 T^{5} + 21935 T^{6} - 1383 p T^{7} + 1214 p^{2} T^{8} - 52 p^{3} T^{9} + 32 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 - 3 T - 3 p T^{2} + 94 T^{3} + 158 p T^{4} + 904 T^{5} - 123467 T^{6} - 308042 T^{7} + 1215550 T^{8} + 10832851 T^{9} + 73693658 T^{10} - 174959016 T^{11} - 3324017493 T^{12} - 174959016 p T^{13} + 73693658 p^{2} T^{14} + 10832851 p^{3} T^{15} + 1215550 p^{4} T^{16} - 308042 p^{5} T^{17} - 123467 p^{6} T^{18} + 904 p^{7} T^{19} + 158 p^{9} T^{20} + 94 p^{9} T^{21} - 3 p^{11} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 16 T + 20 T^{2} + 594 T^{3} + 2163 T^{4} - 43649 T^{5} - 56125 T^{6} + 1282696 T^{7} + 2984747 T^{8} - 22743273 T^{9} - 180497697 T^{10} + 655302586 T^{11} + 2182678017 T^{12} + 655302586 p T^{13} - 180497697 p^{2} T^{14} - 22743273 p^{3} T^{15} + 2984747 p^{4} T^{16} + 1282696 p^{5} T^{17} - 56125 p^{6} T^{18} - 43649 p^{7} T^{19} + 2163 p^{8} T^{20} + 594 p^{9} T^{21} + 20 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( ( 1 - 13 T + 184 T^{2} - 1054 T^{3} + 7158 T^{4} - 10573 T^{5} + 113729 T^{6} - 10573 p T^{7} + 7158 p^{2} T^{8} - 1054 p^{3} T^{9} + 184 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 8 T - 161 T^{2} - 924 T^{3} + 18241 T^{4} + 64367 T^{5} - 1502654 T^{6} - 3175261 T^{7} + 96068491 T^{8} + 105301221 T^{9} - 5078164754 T^{10} - 1647875431 T^{11} + 226350132753 T^{12} - 1647875431 p T^{13} - 5078164754 p^{2} T^{14} + 105301221 p^{3} T^{15} + 96068491 p^{4} T^{16} - 3175261 p^{5} T^{17} - 1502654 p^{6} T^{18} + 64367 p^{7} T^{19} + 18241 p^{8} T^{20} - 924 p^{9} T^{21} - 161 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 11 T - 138 T^{2} - 1349 T^{3} + 16370 T^{4} + 106653 T^{5} - 1472431 T^{6} - 5757651 T^{7} + 106708219 T^{8} + 224797058 T^{9} - 6088028976 T^{10} - 3777766292 T^{11} + 288640495545 T^{12} - 3777766292 p T^{13} - 6088028976 p^{2} T^{14} + 224797058 p^{3} T^{15} + 106708219 p^{4} T^{16} - 5757651 p^{5} T^{17} - 1472431 p^{6} T^{18} + 106653 p^{7} T^{19} + 16370 p^{8} T^{20} - 1349 p^{9} T^{21} - 138 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + T - 104 T^{2} - 189 T^{3} + 5335 T^{4} + 164 p T^{5} - 69863 T^{6} + 514255 T^{7} - 7627520 T^{8} - 55687467 T^{9} + 662939941 T^{10} + 1686387922 T^{11} - 35399065407 T^{12} + 1686387922 p T^{13} + 662939941 p^{2} T^{14} - 55687467 p^{3} T^{15} - 7627520 p^{4} T^{16} + 514255 p^{5} T^{17} - 69863 p^{6} T^{18} + 164 p^{8} T^{19} + 5335 p^{8} T^{20} - 189 p^{9} T^{21} - 104 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 2 T - 214 T^{2} - 252 T^{3} + 24796 T^{4} + 13772 T^{5} - 1921862 T^{6} + 82142 T^{7} + 113089342 T^{8} - 43114584 T^{9} - 5653831794 T^{10} + 1443208718 T^{11} + 285781391787 T^{12} + 1443208718 p T^{13} - 5653831794 p^{2} T^{14} - 43114584 p^{3} T^{15} + 113089342 p^{4} T^{16} + 82142 p^{5} T^{17} - 1921862 p^{6} T^{18} + 13772 p^{7} T^{19} + 24796 p^{8} T^{20} - 252 p^{9} T^{21} - 214 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( ( 1 + 13 T + 5 p T^{2} + 2839 T^{3} + 38957 T^{4} + 294699 T^{5} + 2963017 T^{6} + 294699 p T^{7} + 38957 p^{2} T^{8} + 2839 p^{3} T^{9} + 5 p^{5} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( 1 + 5 T - 140 T^{2} - 373 T^{3} + 8487 T^{4} - 5202 T^{5} - 147441 T^{6} + 963135 T^{7} - 4711566 T^{8} - 13690661 T^{9} - 1296684385 T^{10} - 689962304 T^{11} + 162150963097 T^{12} - 689962304 p T^{13} - 1296684385 p^{2} T^{14} - 13690661 p^{3} T^{15} - 4711566 p^{4} T^{16} + 963135 p^{5} T^{17} - 147441 p^{6} T^{18} - 5202 p^{7} T^{19} + 8487 p^{8} T^{20} - 373 p^{9} T^{21} - 140 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 11 T - 175 T^{2} - 2336 T^{3} + 15663 T^{4} + 247450 T^{5} - 15954 p T^{6} - 18125445 T^{7} + 60512732 T^{8} + 977936543 T^{9} - 2490157221 T^{10} - 26393757979 T^{11} + 95373451231 T^{12} - 26393757979 p T^{13} - 2490157221 p^{2} T^{14} + 977936543 p^{3} T^{15} + 60512732 p^{4} T^{16} - 18125445 p^{5} T^{17} - 15954 p^{7} T^{18} + 247450 p^{7} T^{19} + 15663 p^{8} T^{20} - 2336 p^{9} T^{21} - 175 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 6 T - 249 T^{2} + 278 T^{3} + 39793 T^{4} + 68141 T^{5} - 3761552 T^{6} - 15648583 T^{7} + 241594531 T^{8} + 1275513473 T^{9} - 10122739162 T^{10} - 44683203723 T^{11} + 523547364015 T^{12} - 44683203723 p T^{13} - 10122739162 p^{2} T^{14} + 1275513473 p^{3} T^{15} + 241594531 p^{4} T^{16} - 15648583 p^{5} T^{17} - 3761552 p^{6} T^{18} + 68141 p^{7} T^{19} + 39793 p^{8} T^{20} + 278 p^{9} T^{21} - 249 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 30 T + 224 T^{2} - 1118 T^{3} - 5021 T^{4} + 290169 T^{5} + 1854677 T^{6} - 9817892 T^{7} - 27971653 T^{8} + 688598777 T^{9} - 1819010273 T^{10} - 20701972840 T^{11} + 235631264151 T^{12} - 20701972840 p T^{13} - 1819010273 p^{2} T^{14} + 688598777 p^{3} T^{15} - 27971653 p^{4} T^{16} - 9817892 p^{5} T^{17} + 1854677 p^{6} T^{18} + 290169 p^{7} T^{19} - 5021 p^{8} T^{20} - 1118 p^{9} T^{21} + 224 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 7 T - 277 T^{2} + 2628 T^{3} + 34995 T^{4} - 387429 T^{5} - 3070086 T^{6} + 26237658 T^{7} + 339376855 T^{8} - 565746882 T^{9} - 45365142063 T^{10} - 8895648284 T^{11} + 4474615429807 T^{12} - 8895648284 p T^{13} - 45365142063 p^{2} T^{14} - 565746882 p^{3} T^{15} + 339376855 p^{4} T^{16} + 26237658 p^{5} T^{17} - 3070086 p^{6} T^{18} - 387429 p^{7} T^{19} + 34995 p^{8} T^{20} + 2628 p^{9} T^{21} - 277 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 + 27 T + 656 T^{2} + 10802 T^{3} + 153994 T^{4} + 1760871 T^{5} + 17670883 T^{6} + 1760871 p T^{7} + 153994 p^{2} T^{8} + 10802 p^{3} T^{9} + 656 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( ( 1 + 4 T + 167 T^{2} + 1648 T^{3} + 21035 T^{4} + 202110 T^{5} + 2204075 T^{6} + 202110 p T^{7} + 21035 p^{2} T^{8} + 1648 p^{3} T^{9} + 167 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 + 35 T + 278 T^{2} - 3177 T^{3} - 20496 T^{4} + 1111333 T^{5} + 13328183 T^{6} - 54713297 T^{7} - 1182920923 T^{8} + 11775176076 T^{9} + 251445486222 T^{10} + 186060844192 T^{11} - 17274836413101 T^{12} + 186060844192 p T^{13} + 251445486222 p^{2} T^{14} + 11775176076 p^{3} T^{15} - 1182920923 p^{4} T^{16} - 54713297 p^{5} T^{17} + 13328183 p^{6} T^{18} + 1111333 p^{7} T^{19} - 20496 p^{8} T^{20} - 3177 p^{9} T^{21} + 278 p^{10} T^{22} + 35 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.28070665609173707208846684148, −4.74818788486658011413846152620, −4.59538770372467674838971163489, −4.58230946543916308580631230010, −4.57519299751439993358271528029, −4.55531960880949055981187932399, −4.53259423803456685867788893299, −4.47578615373994744792370653383, −4.44274701910350009850597901794, −4.18697088988592255677401499061, −4.16169913094958830987836304505, −3.93957760638755693920728497407, −3.60551067540227648957971749912, −3.29798952854065220458022770473, −3.22984496892227500874324299375, −3.03834500414905677830030191429, −2.87269774456217944358029465116, −2.75702149941998175409532461712, −2.64012804798322466310884390192, −2.35076883251026167736757838044, −1.72563343605294342870054241683, −1.62808694153426080522566773291, −1.51908533655827400352476466536, −1.20050738829685340082387120652, −0.63091211390540319228885632549, 0.63091211390540319228885632549, 1.20050738829685340082387120652, 1.51908533655827400352476466536, 1.62808694153426080522566773291, 1.72563343605294342870054241683, 2.35076883251026167736757838044, 2.64012804798322466310884390192, 2.75702149941998175409532461712, 2.87269774456217944358029465116, 3.03834500414905677830030191429, 3.22984496892227500874324299375, 3.29798952854065220458022770473, 3.60551067540227648957971749912, 3.93957760638755693920728497407, 4.16169913094958830987836304505, 4.18697088988592255677401499061, 4.44274701910350009850597901794, 4.47578615373994744792370653383, 4.53259423803456685867788893299, 4.55531960880949055981187932399, 4.57519299751439993358271528029, 4.58230946543916308580631230010, 4.59538770372467674838971163489, 4.74818788486658011413846152620, 5.28070665609173707208846684148
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.