Dirichlet series
| L(s) = 1 | − 3.86e3·3-s + 1.68e5·5-s + 4.94e6·7-s − 2.17e7·9-s − 5.80e7·11-s − 1.59e7·13-s − 6.51e8·15-s − 2.87e9·17-s − 3.43e9·19-s − 1.90e10·21-s − 3.45e10·23-s − 6.27e10·25-s + 8.74e10·27-s − 4.64e10·29-s − 3.00e11·31-s + 2.24e11·33-s + 8.32e11·35-s − 5.09e11·37-s + 6.16e10·39-s + 1.61e12·41-s − 2.72e12·43-s − 3.65e12·45-s + 1.56e12·47-s + 1.42e13·49-s + 1.11e13·51-s + 2.61e12·53-s − 9.78e12·55-s + ⋯ |
| L(s) = 1 | − 1.02·3-s + 0.964·5-s + 2.26·7-s − 1.51·9-s − 0.898·11-s − 0.0705·13-s − 0.984·15-s − 1.70·17-s − 0.881·19-s − 2.31·21-s − 2.11·23-s − 2.05·25-s + 1.60·27-s − 0.500·29-s − 1.96·31-s + 0.916·33-s + 2.18·35-s − 0.882·37-s + 0.0719·39-s + 1.29·41-s − 1.53·43-s − 1.45·45-s + 0.451·47-s + 3·49-s + 1.73·51-s + 0.305·53-s − 0.866·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(16-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+15/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{24} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.66622\times 10^{13}\) |
| Root analytic conductor: | \(12.6418\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{24} \cdot 7^{6} ,\ ( \ : [15/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(8)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{17}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 - p^{7} T )^{6} \) | |
| good | 3 | \( 1 + 1288 p T + 4071562 p^{2} T^{2} + 5113076248 p^{3} T^{3} + 7900534326743 p^{4} T^{4} + 3958705525032272 p^{6} T^{5} + 487298793126504100 p^{9} T^{6} + 3958705525032272 p^{21} T^{7} + 7900534326743 p^{34} T^{8} + 5113076248 p^{48} T^{9} + 4071562 p^{62} T^{10} + 1288 p^{76} T^{11} + p^{90} T^{12} \) |
| 5 | \( 1 - 168532 T + 91138104434 T^{2} - 2550182090115268 p T^{3} + 6359229011195923239 p^{4} T^{4} - \)\(40\!\cdots\!12\)\( p^{3} T^{5} + \)\(19\!\cdots\!44\)\( p^{4} T^{6} - \)\(40\!\cdots\!12\)\( p^{18} T^{7} + 6359229011195923239 p^{34} T^{8} - 2550182090115268 p^{46} T^{9} + 91138104434 p^{60} T^{10} - 168532 p^{75} T^{11} + p^{90} T^{12} \) | |
| 11 | \( 1 + 58040072 T + 1007318920054374 p T^{2} + \)\(56\!\cdots\!08\)\( p^{2} T^{3} + \)\(48\!\cdots\!69\)\( p^{3} T^{4} + \)\(24\!\cdots\!32\)\( p^{4} T^{5} + \)\(18\!\cdots\!24\)\( p^{5} T^{6} + \)\(24\!\cdots\!32\)\( p^{19} T^{7} + \)\(48\!\cdots\!69\)\( p^{33} T^{8} + \)\(56\!\cdots\!08\)\( p^{47} T^{9} + 1007318920054374 p^{61} T^{10} + 58040072 p^{75} T^{11} + p^{90} T^{12} \) | |
| 13 | \( 1 + 15956444 T + 257992576023801506 T^{2} + \)\(20\!\cdots\!32\)\( p T^{3} + \)\(17\!\cdots\!91\)\( p^{2} T^{4} + \)\(92\!\cdots\!96\)\( p^{3} T^{5} + \)\(68\!\cdots\!76\)\( p^{4} T^{6} + \)\(92\!\cdots\!96\)\( p^{18} T^{7} + \)\(17\!\cdots\!91\)\( p^{32} T^{8} + \)\(20\!\cdots\!32\)\( p^{46} T^{9} + 257992576023801506 p^{60} T^{10} + 15956444 p^{75} T^{11} + p^{90} T^{12} \) | |
| 17 | \( 1 + 2879979060 T + 12933900526500133122 T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(69\!\cdots\!79\)\( T^{4} + \)\(10\!\cdots\!60\)\( T^{5} + \)\(23\!\cdots\!52\)\( T^{6} + \)\(10\!\cdots\!60\)\( p^{15} T^{7} + \)\(69\!\cdots\!79\)\( p^{30} T^{8} + \)\(24\!\cdots\!80\)\( p^{45} T^{9} + 12933900526500133122 p^{60} T^{10} + 2879979060 p^{75} T^{11} + p^{90} T^{12} \) | |
| 19 | \( 1 + 3435500376 T + 41746558954703750586 T^{2} + \)\(93\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!51\)\( T^{4} + \)\(21\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!92\)\( p^{15} T^{7} + \)\(10\!\cdots\!51\)\( p^{30} T^{8} + \)\(93\!\cdots\!76\)\( p^{45} T^{9} + 41746558954703750586 p^{60} T^{10} + 3435500376 p^{75} T^{11} + p^{90} T^{12} \) | |
| 23 | \( 1 + 34508782592 T + \)\(15\!\cdots\!50\)\( T^{2} + \)\(37\!\cdots\!08\)\( T^{3} + \)\(99\!\cdots\!31\)\( T^{4} + \)\(17\!\cdots\!36\)\( T^{5} + \)\(34\!\cdots\!52\)\( T^{6} + \)\(17\!\cdots\!36\)\( p^{15} T^{7} + \)\(99\!\cdots\!31\)\( p^{30} T^{8} + \)\(37\!\cdots\!08\)\( p^{45} T^{9} + \)\(15\!\cdots\!50\)\( p^{60} T^{10} + 34508782592 p^{75} T^{11} + p^{90} T^{12} \) | |
| 29 | \( 1 + 46458543420 T + \)\(66\!\cdots\!70\)\( T^{2} - \)\(91\!\cdots\!32\)\( T^{3} + \)\(14\!\cdots\!03\)\( T^{4} + \)\(24\!\cdots\!32\)\( T^{5} + \)\(14\!\cdots\!60\)\( T^{6} + \)\(24\!\cdots\!32\)\( p^{15} T^{7} + \)\(14\!\cdots\!03\)\( p^{30} T^{8} - \)\(91\!\cdots\!32\)\( p^{45} T^{9} + \)\(66\!\cdots\!70\)\( p^{60} T^{10} + 46458543420 p^{75} T^{11} + p^{90} T^{12} \) | |
| 31 | \( 1 + 300765178336 T + \)\(68\!\cdots\!90\)\( T^{2} + \)\(12\!\cdots\!28\)\( T^{3} + \)\(14\!\cdots\!19\)\( T^{4} + \)\(14\!\cdots\!84\)\( T^{5} + \)\(17\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!84\)\( p^{15} T^{7} + \)\(14\!\cdots\!19\)\( p^{30} T^{8} + \)\(12\!\cdots\!28\)\( p^{45} T^{9} + \)\(68\!\cdots\!90\)\( p^{60} T^{10} + 300765178336 p^{75} T^{11} + p^{90} T^{12} \) | |
| 37 | \( 1 + 509722422412 T + \)\(12\!\cdots\!10\)\( T^{2} + \)\(42\!\cdots\!64\)\( T^{3} + \)\(69\!\cdots\!27\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!00\)\( p^{15} T^{7} + \)\(69\!\cdots\!27\)\( p^{30} T^{8} + \)\(42\!\cdots\!64\)\( p^{45} T^{9} + \)\(12\!\cdots\!10\)\( p^{60} T^{10} + 509722422412 p^{75} T^{11} + p^{90} T^{12} \) | |
| 41 | \( 1 - 1619276439676 T + \)\(69\!\cdots\!62\)\( T^{2} - \)\(10\!\cdots\!52\)\( T^{3} + \)\(24\!\cdots\!99\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(48\!\cdots\!84\)\( T^{6} - \)\(29\!\cdots\!76\)\( p^{15} T^{7} + \)\(24\!\cdots\!99\)\( p^{30} T^{8} - \)\(10\!\cdots\!52\)\( p^{45} T^{9} + \)\(69\!\cdots\!62\)\( p^{60} T^{10} - 1619276439676 p^{75} T^{11} + p^{90} T^{12} \) | |
| 43 | \( 1 + 2727781008680 T + \)\(96\!\cdots\!46\)\( T^{2} + \)\(12\!\cdots\!16\)\( T^{3} + \)\(28\!\cdots\!07\)\( T^{4} + \)\(12\!\cdots\!84\)\( T^{5} + \)\(55\!\cdots\!24\)\( T^{6} + \)\(12\!\cdots\!84\)\( p^{15} T^{7} + \)\(28\!\cdots\!07\)\( p^{30} T^{8} + \)\(12\!\cdots\!16\)\( p^{45} T^{9} + \)\(96\!\cdots\!46\)\( p^{60} T^{10} + 2727781008680 p^{75} T^{11} + p^{90} T^{12} \) | |
| 47 | \( 1 - 1567726443808 T - \)\(81\!\cdots\!94\)\( T^{2} - \)\(10\!\cdots\!36\)\( T^{3} + \)\(35\!\cdots\!67\)\( T^{4} - \)\(30\!\cdots\!00\)\( T^{5} - \)\(19\!\cdots\!64\)\( T^{6} - \)\(30\!\cdots\!00\)\( p^{15} T^{7} + \)\(35\!\cdots\!67\)\( p^{30} T^{8} - \)\(10\!\cdots\!36\)\( p^{45} T^{9} - \)\(81\!\cdots\!94\)\( p^{60} T^{10} - 1567726443808 p^{75} T^{11} + p^{90} T^{12} \) | |
| 53 | \( 1 - 2612518893588 T + \)\(23\!\cdots\!94\)\( T^{2} - \)\(73\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!51\)\( T^{4} - \)\(76\!\cdots\!56\)\( T^{5} + \)\(22\!\cdots\!80\)\( T^{6} - \)\(76\!\cdots\!56\)\( p^{15} T^{7} + \)\(27\!\cdots\!51\)\( p^{30} T^{8} - \)\(73\!\cdots\!20\)\( p^{45} T^{9} + \)\(23\!\cdots\!94\)\( p^{60} T^{10} - 2612518893588 p^{75} T^{11} + p^{90} T^{12} \) | |
| 59 | \( 1 + 5112561397384 T - \)\(34\!\cdots\!22\)\( T^{2} - \)\(17\!\cdots\!44\)\( T^{3} + \)\(21\!\cdots\!11\)\( T^{4} + \)\(92\!\cdots\!08\)\( T^{5} + \)\(15\!\cdots\!72\)\( T^{6} + \)\(92\!\cdots\!08\)\( p^{15} T^{7} + \)\(21\!\cdots\!11\)\( p^{30} T^{8} - \)\(17\!\cdots\!44\)\( p^{45} T^{9} - \)\(34\!\cdots\!22\)\( p^{60} T^{10} + 5112561397384 p^{75} T^{11} + p^{90} T^{12} \) | |
| 61 | \( 1 - 4545346033732 T + \)\(26\!\cdots\!54\)\( T^{2} - \)\(18\!\cdots\!12\)\( T^{3} + \)\(33\!\cdots\!15\)\( T^{4} - \)\(24\!\cdots\!96\)\( T^{5} + \)\(25\!\cdots\!24\)\( T^{6} - \)\(24\!\cdots\!96\)\( p^{15} T^{7} + \)\(33\!\cdots\!15\)\( p^{30} T^{8} - \)\(18\!\cdots\!12\)\( p^{45} T^{9} + \)\(26\!\cdots\!54\)\( p^{60} T^{10} - 4545346033732 p^{75} T^{11} + p^{90} T^{12} \) | |
| 67 | \( 1 + 74088461612408 T + \)\(11\!\cdots\!06\)\( T^{2} + \)\(47\!\cdots\!32\)\( T^{3} + \)\(45\!\cdots\!15\)\( T^{4} + \)\(13\!\cdots\!68\)\( T^{5} + \)\(12\!\cdots\!40\)\( T^{6} + \)\(13\!\cdots\!68\)\( p^{15} T^{7} + \)\(45\!\cdots\!15\)\( p^{30} T^{8} + \)\(47\!\cdots\!32\)\( p^{45} T^{9} + \)\(11\!\cdots\!06\)\( p^{60} T^{10} + 74088461612408 p^{75} T^{11} + p^{90} T^{12} \) | |
| 71 | \( 1 + 51092191255952 T + \)\(24\!\cdots\!58\)\( T^{2} + \)\(76\!\cdots\!56\)\( T^{3} + \)\(25\!\cdots\!91\)\( T^{4} + \)\(47\!\cdots\!08\)\( T^{5} + \)\(16\!\cdots\!68\)\( T^{6} + \)\(47\!\cdots\!08\)\( p^{15} T^{7} + \)\(25\!\cdots\!91\)\( p^{30} T^{8} + \)\(76\!\cdots\!56\)\( p^{45} T^{9} + \)\(24\!\cdots\!58\)\( p^{60} T^{10} + 51092191255952 p^{75} T^{11} + p^{90} T^{12} \) | |
| 73 | \( 1 - 294343628983292 T + \)\(60\!\cdots\!86\)\( T^{2} - \)\(83\!\cdots\!36\)\( T^{3} + \)\(99\!\cdots\!39\)\( T^{4} - \)\(10\!\cdots\!24\)\( T^{5} + \)\(98\!\cdots\!28\)\( T^{6} - \)\(10\!\cdots\!24\)\( p^{15} T^{7} + \)\(99\!\cdots\!39\)\( p^{30} T^{8} - \)\(83\!\cdots\!36\)\( p^{45} T^{9} + \)\(60\!\cdots\!86\)\( p^{60} T^{10} - 294343628983292 p^{75} T^{11} + p^{90} T^{12} \) | |
| 79 | \( 1 + 56070305620928 T + \)\(11\!\cdots\!98\)\( T^{2} + \)\(78\!\cdots\!44\)\( T^{3} + \)\(66\!\cdots\!71\)\( T^{4} + \)\(40\!\cdots\!80\)\( T^{5} + \)\(23\!\cdots\!44\)\( T^{6} + \)\(40\!\cdots\!80\)\( p^{15} T^{7} + \)\(66\!\cdots\!71\)\( p^{30} T^{8} + \)\(78\!\cdots\!44\)\( p^{45} T^{9} + \)\(11\!\cdots\!98\)\( p^{60} T^{10} + 56070305620928 p^{75} T^{11} + p^{90} T^{12} \) | |
| 83 | \( 1 + 984559173278520 T + \)\(62\!\cdots\!02\)\( T^{2} + \)\(28\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!67\)\( T^{4} + \)\(32\!\cdots\!80\)\( T^{5} + \)\(86\!\cdots\!40\)\( T^{6} + \)\(32\!\cdots\!80\)\( p^{15} T^{7} + \)\(10\!\cdots\!67\)\( p^{30} T^{8} + \)\(28\!\cdots\!60\)\( p^{45} T^{9} + \)\(62\!\cdots\!02\)\( p^{60} T^{10} + 984559173278520 p^{75} T^{11} + p^{90} T^{12} \) | |
| 89 | \( 1 - 1528383063352572 T + \)\(17\!\cdots\!98\)\( T^{2} - \)\(13\!\cdots\!08\)\( T^{3} + \)\(89\!\cdots\!99\)\( T^{4} - \)\(47\!\cdots\!60\)\( T^{5} + \)\(21\!\cdots\!40\)\( T^{6} - \)\(47\!\cdots\!60\)\( p^{15} T^{7} + \)\(89\!\cdots\!99\)\( p^{30} T^{8} - \)\(13\!\cdots\!08\)\( p^{45} T^{9} + \)\(17\!\cdots\!98\)\( p^{60} T^{10} - 1528383063352572 p^{75} T^{11} + p^{90} T^{12} \) | |
| 97 | \( 1 - 2255703641701580 T + \)\(41\!\cdots\!42\)\( T^{2} - \)\(43\!\cdots\!60\)\( T^{3} + \)\(41\!\cdots\!63\)\( T^{4} - \)\(28\!\cdots\!20\)\( T^{5} + \)\(23\!\cdots\!88\)\( T^{6} - \)\(28\!\cdots\!20\)\( p^{15} T^{7} + \)\(41\!\cdots\!63\)\( p^{30} T^{8} - \)\(43\!\cdots\!60\)\( p^{45} T^{9} + \)\(41\!\cdots\!42\)\( p^{60} T^{10} - 2255703641701580 p^{75} T^{11} + p^{90} T^{12} \) | |
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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
−5.72487754056068986208354225783, −5.33150890821452509547087879079, −5.02463782244852490043483366606, −4.93544059314731875922052427726, −4.75660751122110057214503021678, −4.69510474136104724421915418187, −4.68736610738028797818084387364, −4.14283017598192127803204536880, −3.84231730070537457314490471232, −3.79985378215024319467561837923, −3.59503049673040953107485139224, −3.46343938527021099369893917309, −3.43244687315974842666031051127, −2.59286392746440702960531868160, −2.54682569440899792854915205092, −2.46509268212089673949158087106, −2.19275431901911406176968064816, −2.17107019332759911697902716307, −2.05891741236454714982041665371, −1.94484833684366995152061414484, −1.49241519895319848044275534113, −1.39313749122976759106749822184, −1.13533116352155130950403462198, −1.04728567319217160375318849225, −0.75444554297117310304483152578, 0, 0, 0, 0, 0, 0, 0.75444554297117310304483152578, 1.04728567319217160375318849225, 1.13533116352155130950403462198, 1.39313749122976759106749822184, 1.49241519895319848044275534113, 1.94484833684366995152061414484, 2.05891741236454714982041665371, 2.17107019332759911697902716307, 2.19275431901911406176968064816, 2.46509268212089673949158087106, 2.54682569440899792854915205092, 2.59286392746440702960531868160, 3.43244687315974842666031051127, 3.46343938527021099369893917309, 3.59503049673040953107485139224, 3.79985378215024319467561837923, 3.84231730070537457314490471232, 4.14283017598192127803204536880, 4.68736610738028797818084387364, 4.69510474136104724421915418187, 4.75660751122110057214503021678, 4.93544059314731875922052427726, 5.02463782244852490043483366606, 5.33150890821452509547087879079, 5.72487754056068986208354225783
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.