Dirichlet series
| L(s) = 1 | − 2-s + 6·3-s − 17·4-s − 6·6-s − 4·7-s + 25·8-s − 69·9-s − 20·11-s − 102·12-s − 42·13-s + 4·14-s + 78·16-s − 102·17-s + 69·18-s − 70·19-s − 24·21-s + 20·22-s + 92·23-s + 150·24-s + 42·26-s − 414·27-s + 68·28-s − 316·29-s − 758·31-s − 174·32-s − 120·33-s + 102·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.15·3-s − 2.12·4-s − 0.408·6-s − 0.215·7-s + 1.10·8-s − 2.55·9-s − 0.548·11-s − 2.45·12-s − 0.896·13-s + 0.0763·14-s + 1.21·16-s − 1.45·17-s + 0.903·18-s − 0.845·19-s − 0.249·21-s + 0.193·22-s + 0.834·23-s + 1.27·24-s + 0.316·26-s − 2.95·27-s + 0.458·28-s − 2.02·29-s − 4.39·31-s − 0.961·32-s − 0.633·33-s + 0.514·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(5^{12} \cdot 17^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.48616\times 10^{8}\) |
| Root analytic conductor: | \(5.00757\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 5^{12} \cdot 17^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 17 | \( ( 1 + p T )^{6} \) | |
| good | 2 | \( 1 + T + 9 p T^{2} + 5 p T^{3} + 213 T^{4} + 29 T^{5} + 463 p^{2} T^{6} + 29 p^{3} T^{7} + 213 p^{6} T^{8} + 5 p^{10} T^{9} + 9 p^{13} T^{10} + p^{15} T^{11} + p^{18} T^{12} \) |
| 3 | \( 1 - 2 p T + 35 p T^{2} - 70 p^{2} T^{3} + 1904 p T^{4} - 10144 p T^{5} + 189869 T^{6} - 10144 p^{4} T^{7} + 1904 p^{7} T^{8} - 70 p^{11} T^{9} + 35 p^{13} T^{10} - 2 p^{16} T^{11} + p^{18} T^{12} \) | |
| 7 | \( 1 + 4 T + 1079 T^{2} + 1168 T^{3} + 94062 p T^{4} + 98386 T^{5} + 266269953 T^{6} + 98386 p^{3} T^{7} + 94062 p^{7} T^{8} + 1168 p^{9} T^{9} + 1079 p^{12} T^{10} + 4 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 + 20 T + 4948 T^{2} + 43012 T^{3} + 10814227 T^{4} + 11855292 T^{5} + 15873719200 T^{6} + 11855292 p^{3} T^{7} + 10814227 p^{6} T^{8} + 43012 p^{9} T^{9} + 4948 p^{12} T^{10} + 20 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 + 42 T + 7869 T^{2} + 151042 T^{3} + 24727174 T^{4} + 111870518 T^{5} + 54714903749 T^{6} + 111870518 p^{3} T^{7} + 24727174 p^{6} T^{8} + 151042 p^{9} T^{9} + 7869 p^{12} T^{10} + 42 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 + 70 T + 26826 T^{2} + 2464698 T^{3} + 330431959 T^{4} + 34215767068 T^{5} + 2653157592748 T^{6} + 34215767068 p^{3} T^{7} + 330431959 p^{6} T^{8} + 2464698 p^{9} T^{9} + 26826 p^{12} T^{10} + 70 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 - 4 p T + 42788 T^{2} - 5436848 T^{3} + 830575523 T^{4} - 129470295824 T^{5} + 11130729824624 T^{6} - 129470295824 p^{3} T^{7} + 830575523 p^{6} T^{8} - 5436848 p^{9} T^{9} + 42788 p^{12} T^{10} - 4 p^{16} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 316 T + 109526 T^{2} + 23100548 T^{3} + 4747718199 T^{4} + 823410852832 T^{5} + 133168996602228 T^{6} + 823410852832 p^{3} T^{7} + 4747718199 p^{6} T^{8} + 23100548 p^{9} T^{9} + 109526 p^{12} T^{10} + 316 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 + 758 T + 360977 T^{2} + 123118754 T^{3} + 33543209988 T^{4} + 7472355028736 T^{5} + 1406393966922489 T^{6} + 7472355028736 p^{3} T^{7} + 33543209988 p^{6} T^{8} + 123118754 p^{9} T^{9} + 360977 p^{12} T^{10} + 758 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 + 76 T + 169906 T^{2} + 10502668 T^{3} + 15896846071 T^{4} + 745784974728 T^{5} + 956093792902748 T^{6} + 745784974728 p^{3} T^{7} + 15896846071 p^{6} T^{8} + 10502668 p^{9} T^{9} + 169906 p^{12} T^{10} + 76 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 + 512 T + 364466 T^{2} + 138040792 T^{3} + 58919063775 T^{4} + 17026052439928 T^{5} + 5298258739038460 T^{6} + 17026052439928 p^{3} T^{7} + 58919063775 p^{6} T^{8} + 138040792 p^{9} T^{9} + 364466 p^{12} T^{10} + 512 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 + 70 T + 197874 T^{2} - 29168326 T^{3} + 10826853927 T^{4} - 7592990627284 T^{5} + 302366159104508 T^{6} - 7592990627284 p^{3} T^{7} + 10826853927 p^{6} T^{8} - 29168326 p^{9} T^{9} + 197874 p^{12} T^{10} + 70 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 - 448 T + 580798 T^{2} - 198993872 T^{3} + 141567396175 T^{4} - 37888800109264 T^{5} + 19115530685467620 T^{6} - 37888800109264 p^{3} T^{7} + 141567396175 p^{6} T^{8} - 198993872 p^{9} T^{9} + 580798 p^{12} T^{10} - 448 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 734 T + 847989 T^{2} + 431012234 T^{3} + 287021168218 T^{4} + 110583765819462 T^{5} + 54487722665556021 T^{6} + 110583765819462 p^{3} T^{7} + 287021168218 p^{6} T^{8} + 431012234 p^{9} T^{9} + 847989 p^{12} T^{10} + 734 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 238 T + 995334 T^{2} + 197942850 T^{3} + 452027119815 T^{4} + 73669147883468 T^{5} + 118899853181909524 T^{6} + 73669147883468 p^{3} T^{7} + 452027119815 p^{6} T^{8} + 197942850 p^{9} T^{9} + 995334 p^{12} T^{10} + 238 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 + 1188 T + 1748090 T^{2} + 1367574628 T^{3} + 1116299329335 T^{4} + 624612049275464 T^{5} + 350026644826364172 T^{6} + 624612049275464 p^{3} T^{7} + 1116299329335 p^{6} T^{8} + 1367574628 p^{9} T^{9} + 1748090 p^{12} T^{10} + 1188 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 768 T + 706690 T^{2} - 192601280 T^{3} + 237734438775 T^{4} - 127111011778624 T^{5} + 127125428249077628 T^{6} - 127111011778624 p^{3} T^{7} + 237734438775 p^{6} T^{8} - 192601280 p^{9} T^{9} + 706690 p^{12} T^{10} - 768 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 1276 T + 1911811 T^{2} + 1562133684 T^{3} + 1375832897030 T^{4} + 868606251808150 T^{5} + 592866246651136721 T^{6} + 868606251808150 p^{3} T^{7} + 1375832897030 p^{6} T^{8} + 1562133684 p^{9} T^{9} + 1911811 p^{12} T^{10} + 1276 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 + 84 T + 1019274 T^{2} + 8385356 T^{3} + 625009606863 T^{4} + 32327144607888 T^{5} + 300051831193873292 T^{6} + 32327144607888 p^{3} T^{7} + 625009606863 p^{6} T^{8} + 8385356 p^{9} T^{9} + 1019274 p^{12} T^{10} + 84 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 3066 T + 5975027 T^{2} + 8147234562 T^{3} + 8883522900366 T^{4} + 7915792843234616 T^{5} + 6033024976452237145 T^{6} + 7915792843234616 p^{3} T^{7} + 8883522900366 p^{6} T^{8} + 8147234562 p^{9} T^{9} + 5975027 p^{12} T^{10} + 3066 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 + 92 T + 1323682 T^{2} + 273166884 T^{3} + 1286946563863 T^{4} + 191484652785656 T^{5} + 860607993399136316 T^{6} + 191484652785656 p^{3} T^{7} + 1286946563863 p^{6} T^{8} + 273166884 p^{9} T^{9} + 1323682 p^{12} T^{10} + 92 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 1760 T + 2798174 T^{2} + 4169476384 T^{3} + 4526987039919 T^{4} + 4570352781182384 T^{5} + 4300812649730173092 T^{6} + 4570352781182384 p^{3} T^{7} + 4526987039919 p^{6} T^{8} + 4169476384 p^{9} T^{9} + 2798174 p^{12} T^{10} + 1760 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 12 p T + 2791838 T^{2} - 3926683212 T^{3} + 5091862013775 T^{4} - 5432759795084648 T^{5} + 6177688080875556580 T^{6} - 5432759795084648 p^{3} T^{7} + 5091862013775 p^{6} T^{8} - 3926683212 p^{9} T^{9} + 2791838 p^{12} T^{10} - 12 p^{16} T^{11} + p^{18} T^{12} \) | |
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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
−6.22613386591412461296648561359, −5.88109819171209603335669000030, −5.57335889864923687374869103025, −5.38713727173116145507439401055, −5.30207028857165183710073217305, −5.25944823858072719605304131250, −5.24294161077458733030047651285, −4.89896620513056292980913755419, −4.56601367235211404810101891550, −4.40873618677941242235883295377, −4.33591653580664017420712198753, −4.04939618099608166281072170109, −4.03643332916246655663454454925, −3.55078489590976208503364549774, −3.46339446809314426930640943390, −3.20570749486415411899448185612, −3.10865382454852942656713926570, −2.88708930413310177293794103579, −2.68398862508619512758105295027, −2.34913138908517530977372910233, −2.31717791152293715593995312592, −1.93149043706285290329073491493, −1.48907858398747149608729112713, −1.48593948131549558195888611335, −1.28840677838444840479038845230, 0, 0, 0, 0, 0, 0, 1.28840677838444840479038845230, 1.48593948131549558195888611335, 1.48907858398747149608729112713, 1.93149043706285290329073491493, 2.31717791152293715593995312592, 2.34913138908517530977372910233, 2.68398862508619512758105295027, 2.88708930413310177293794103579, 3.10865382454852942656713926570, 3.20570749486415411899448185612, 3.46339446809314426930640943390, 3.55078489590976208503364549774, 4.03643332916246655663454454925, 4.04939618099608166281072170109, 4.33591653580664017420712198753, 4.40873618677941242235883295377, 4.56601367235211404810101891550, 4.89896620513056292980913755419, 5.24294161077458733030047651285, 5.25944823858072719605304131250, 5.30207028857165183710073217305, 5.38713727173116145507439401055, 5.57335889864923687374869103025, 5.88109819171209603335669000030, 6.22613386591412461296648561359
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.