Dirichlet series
| L(s) = 1 | + 12·2-s + 18·3-s + 84·4-s + 20·5-s + 216·6-s + 42·7-s + 448·8-s + 189·9-s + 240·10-s + 79·11-s + 1.51e3·12-s + 52·13-s + 504·14-s + 360·15-s + 2.01e3·16-s + 140·17-s + 2.26e3·18-s + 93·19-s + 1.68e3·20-s + 756·21-s + 948·22-s + 138·23-s + 8.06e3·24-s − 104·25-s + 624·26-s + 1.51e3·27-s + 3.52e3·28-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 3.46·3-s + 21/2·4-s + 1.78·5-s + 14.6·6-s + 2.26·7-s + 19.7·8-s + 7·9-s + 7.58·10-s + 2.16·11-s + 36.3·12-s + 1.10·13-s + 9.62·14-s + 6.19·15-s + 63/2·16-s + 1.99·17-s + 29.6·18-s + 1.12·19-s + 18.7·20-s + 7.85·21-s + 9.18·22-s + 1.25·23-s + 68.5·24-s − 0.831·25-s + 4.70·26-s + 10.7·27-s + 23.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.42814\times 10^{10}\) |
| Root analytic conductor: | \(7.54955\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(15065.71773\) |
| \(L(\frac12)\) | \(\approx\) | \(15065.71773\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T )^{6} \) |
| 3 | \( ( 1 - p T )^{6} \) | |
| 7 | \( ( 1 - p T )^{6} \) | |
| 23 | \( ( 1 - p T )^{6} \) | |
| good | 5 | \( 1 - 4 p T + 504 T^{2} - 6547 T^{3} + 112837 T^{4} - 1145009 T^{5} + 15979292 T^{6} - 1145009 p^{3} T^{7} + 112837 p^{6} T^{8} - 6547 p^{9} T^{9} + 504 p^{12} T^{10} - 4 p^{16} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 79 T + 6711 T^{2} - 369082 T^{3} + 19264917 T^{4} - 801756679 T^{5} + 2946721634 p T^{6} - 801756679 p^{3} T^{7} + 19264917 p^{6} T^{8} - 369082 p^{9} T^{9} + 6711 p^{12} T^{10} - 79 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 - 4 p T + 7642 T^{2} - 291085 T^{3} + 26089779 T^{4} - 709747419 T^{5} + 61934073108 T^{6} - 709747419 p^{3} T^{7} + 26089779 p^{6} T^{8} - 291085 p^{9} T^{9} + 7642 p^{12} T^{10} - 4 p^{16} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 - 140 T + 27855 T^{2} - 2755798 T^{3} + 318177331 T^{4} - 23914971206 T^{5} + 2022307747274 T^{6} - 23914971206 p^{3} T^{7} + 318177331 p^{6} T^{8} - 2755798 p^{9} T^{9} + 27855 p^{12} T^{10} - 140 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 - 93 T + 32429 T^{2} - 2531146 T^{3} + 480690119 T^{4} - 30696232061 T^{5} + 4185046434662 T^{6} - 30696232061 p^{3} T^{7} + 480690119 p^{6} T^{8} - 2531146 p^{9} T^{9} + 32429 p^{12} T^{10} - 93 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 - 143 T + 36151 T^{2} - 44736 T^{3} + 729116989 T^{4} - 109434495609 T^{5} + 40817421574854 T^{6} - 109434495609 p^{3} T^{7} + 729116989 p^{6} T^{8} - 44736 p^{9} T^{9} + 36151 p^{12} T^{10} - 143 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 130 T + 128189 T^{2} - 11998040 T^{3} + 7184758791 T^{4} - 509827784166 T^{5} + 253744836972342 T^{6} - 509827784166 p^{3} T^{7} + 7184758791 p^{6} T^{8} - 11998040 p^{9} T^{9} + 128189 p^{12} T^{10} - 130 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 - 151 T + 188165 T^{2} - 22525580 T^{3} + 17394982227 T^{4} - 1738447096101 T^{5} + 1062225084328398 T^{6} - 1738447096101 p^{3} T^{7} + 17394982227 p^{6} T^{8} - 22525580 p^{9} T^{9} + 188165 p^{12} T^{10} - 151 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 - 412 T + 355348 T^{2} - 110892564 T^{3} + 54790130500 T^{4} - 13323488224332 T^{5} + 4826883453712038 T^{6} - 13323488224332 p^{3} T^{7} + 54790130500 p^{6} T^{8} - 110892564 p^{9} T^{9} + 355348 p^{12} T^{10} - 412 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 - 250 T + 225898 T^{2} - 76764731 T^{3} + 32124511489 T^{4} - 8758560311089 T^{5} + 3301560120478280 T^{6} - 8758560311089 p^{3} T^{7} + 32124511489 p^{6} T^{8} - 76764731 p^{9} T^{9} + 225898 p^{12} T^{10} - 250 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 - 666 T + 466143 T^{2} - 192425346 T^{3} + 92225707995 T^{4} - 30541592955168 T^{5} + 11591548136101834 T^{6} - 30541592955168 p^{3} T^{7} + 92225707995 p^{6} T^{8} - 192425346 p^{9} T^{9} + 466143 p^{12} T^{10} - 666 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 96 T + 272976 T^{2} - 7288443 T^{3} + 61149282885 T^{4} - 1976314323405 T^{5} + 10498802066897548 T^{6} - 1976314323405 p^{3} T^{7} + 61149282885 p^{6} T^{8} - 7288443 p^{9} T^{9} + 272976 p^{12} T^{10} + 96 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 - 514 T + 1030590 T^{2} - 424620193 T^{3} + 479685826629 T^{4} - 158364055062727 T^{5} + 127185371458761400 T^{6} - 158364055062727 p^{3} T^{7} + 479685826629 p^{6} T^{8} - 424620193 p^{9} T^{9} + 1030590 p^{12} T^{10} - 514 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 422 T + 705108 T^{2} - 291613603 T^{3} + 264660322647 T^{4} - 86007393245699 T^{5} + 70847461102237256 T^{6} - 86007393245699 p^{3} T^{7} + 264660322647 p^{6} T^{8} - 291613603 p^{9} T^{9} + 705108 p^{12} T^{10} - 422 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 669 T + 232415 T^{2} + 35670863 T^{3} + 7168308395 T^{4} + 2056696429480 T^{5} + 11574723652915658 T^{6} + 2056696429480 p^{3} T^{7} + 7168308395 p^{6} T^{8} + 35670863 p^{9} T^{9} + 232415 p^{12} T^{10} - 669 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 357 T + 456503 T^{2} + 439885621 T^{3} + 209710955995 T^{4} + 127332104830112 T^{5} + 125171187294487066 T^{6} + 127332104830112 p^{3} T^{7} + 209710955995 p^{6} T^{8} + 439885621 p^{9} T^{9} + 456503 p^{12} T^{10} + 357 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 430 T + 1446559 T^{2} + 65309258 T^{3} + 553071721479 T^{4} + 458311887262212 T^{5} + 106592418595920786 T^{6} + 458311887262212 p^{3} T^{7} + 553071721479 p^{6} T^{8} + 65309258 p^{9} T^{9} + 1446559 p^{12} T^{10} - 430 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 - 750 T + 1622513 T^{2} - 799228344 T^{3} + 1234701216467 T^{4} - 504055552651890 T^{5} + 700639267844808566 T^{6} - 504055552651890 p^{3} T^{7} + 1234701216467 p^{6} T^{8} - 799228344 p^{9} T^{9} + 1622513 p^{12} T^{10} - 750 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 222 T + 1544105 T^{2} - 359608964 T^{3} + 1645958205223 T^{4} - 327480804988438 T^{5} + 1066734301262184478 T^{6} - 327480804988438 p^{3} T^{7} + 1645958205223 p^{6} T^{8} - 359608964 p^{9} T^{9} + 1544105 p^{12} T^{10} - 222 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 - 763 T + 2709459 T^{2} - 1458471541 T^{3} + 3487212312255 T^{4} - 1405507577422264 T^{5} + 2873512000389566554 T^{6} - 1405507577422264 p^{3} T^{7} + 3487212312255 p^{6} T^{8} - 1458471541 p^{9} T^{9} + 2709459 p^{12} T^{10} - 763 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 575 T + 807149 T^{2} - 548526844 T^{3} + 2044952421687 T^{4} - 862722130072965 T^{5} + 1056594221113595190 T^{6} - 862722130072965 p^{3} T^{7} + 2044952421687 p^{6} T^{8} - 548526844 p^{9} T^{9} + 807149 p^{12} T^{10} - 575 p^{15} T^{11} + p^{18} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.03655672223843700571782530715, −4.42766475015870370189503749449, −4.39508399340713750163660585766, −4.24202910003682265311604088421, −4.13902278938742739287620084070, −4.13364595924487138667082713342, −4.07817599443140178222001004175, −3.55918287161558882847939056570, −3.44420251936257816391806685866, −3.41976155604049413836578512893, −3.39483253635467080064777367758, −3.16433005154586950767634701360, −3.00923458369706983790581735869, −2.39205668857060470086388128165, −2.37894465078385957815698604079, −2.33266168614738398006703694195, −2.27947616098872259289676308286, −2.26487733769098236735197582854, −1.97805038954060742095264693781, −1.46519063865653760426850393793, −1.26501823073020756578687736316, −1.25241201130438730685037599577, −1.11403954895499479363209576639, −1.04024050043123796864273184725, −0.903246688199711601977518909663, 0.903246688199711601977518909663, 1.04024050043123796864273184725, 1.11403954895499479363209576639, 1.25241201130438730685037599577, 1.26501823073020756578687736316, 1.46519063865653760426850393793, 1.97805038954060742095264693781, 2.26487733769098236735197582854, 2.27947616098872259289676308286, 2.33266168614738398006703694195, 2.37894465078385957815698604079, 2.39205668857060470086388128165, 3.00923458369706983790581735869, 3.16433005154586950767634701360, 3.39483253635467080064777367758, 3.41976155604049413836578512893, 3.44420251936257816391806685866, 3.55918287161558882847939056570, 4.07817599443140178222001004175, 4.13364595924487138667082713342, 4.13902278938742739287620084070, 4.24202910003682265311604088421, 4.39508399340713750163660585766, 4.42766475015870370189503749449, 5.03655672223843700571782530715
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.