Dirichlet series
| L(s) = 1 | + 48·2-s + 162·3-s + 1.34e3·4-s + 332·5-s + 7.77e3·6-s + 2.05e3·7-s + 2.86e4·8-s + 1.53e4·9-s + 1.59e4·10-s + 7.98e3·11-s + 2.17e5·12-s + 1.60e4·13-s + 9.87e4·14-s + 5.37e4·15-s + 5.16e5·16-s + 2.41e4·17-s + 7.34e5·18-s + 4.61e4·19-s + 4.46e5·20-s + 3.33e5·21-s + 3.83e5·22-s + 8.71e3·23-s + 4.64e6·24-s − 8.89e4·25-s + 7.68e5·26-s + 1.10e6·27-s + 2.76e6·28-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 3.46·3-s + 21/2·4-s + 1.18·5-s + 14.6·6-s + 2.26·7-s + 19.7·8-s + 7·9-s + 5.03·10-s + 1.80·11-s + 36.3·12-s + 2.02·13-s + 9.62·14-s + 4.11·15-s + 63/2·16-s + 1.19·17-s + 29.6·18-s + 1.54·19-s + 12.4·20-s + 7.85·21-s + 7.67·22-s + 0.149·23-s + 68.5·24-s − 1.13·25-s + 8.57·26-s + 10.7·27-s + 23.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.03635\times 10^{12}\) |
| Root analytic conductor: | \(12.0134\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(30895.04829\) |
| \(L(\frac12)\) | \(\approx\) | \(30895.04829\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{3} T )^{6} \) |
| 3 | \( ( 1 - p^{3} T )^{6} \) | |
| 7 | \( ( 1 - p^{3} T )^{6} \) | |
| 11 | \( ( 1 - p^{3} T )^{6} \) | |
| good | 5 | \( 1 - 332 T + 199151 T^{2} - 9816414 p T^{3} + 672863711 p^{2} T^{4} - 34440860926 p^{3} T^{5} + 1866564769298 p^{4} T^{6} - 34440860926 p^{10} T^{7} + 672863711 p^{16} T^{8} - 9816414 p^{22} T^{9} + 199151 p^{28} T^{10} - 332 p^{35} T^{11} + p^{42} T^{12} \) |
| 13 | \( 1 - 16014 T + 291175499 T^{2} - 3516889291862 T^{3} + 41518243859249523 T^{4} - \)\(38\!\cdots\!64\)\( T^{5} + \)\(33\!\cdots\!14\)\( T^{6} - \)\(38\!\cdots\!64\)\( p^{7} T^{7} + 41518243859249523 p^{14} T^{8} - 3516889291862 p^{21} T^{9} + 291175499 p^{28} T^{10} - 16014 p^{35} T^{11} + p^{42} T^{12} \) | |
| 17 | \( 1 - 1420 p T + 1627846250 T^{2} - 28398715790916 T^{3} + 1128009566453365439 T^{4} - \)\(15\!\cdots\!52\)\( T^{5} + \)\(51\!\cdots\!36\)\( T^{6} - \)\(15\!\cdots\!52\)\( p^{7} T^{7} + 1128009566453365439 p^{14} T^{8} - 28398715790916 p^{21} T^{9} + 1627846250 p^{28} T^{10} - 1420 p^{36} T^{11} + p^{42} T^{12} \) | |
| 19 | \( 1 - 46190 T + 196792867 p T^{2} - 129394076389776 T^{3} + 6554166746730256755 T^{4} - \)\(18\!\cdots\!22\)\( T^{5} + \)\(73\!\cdots\!62\)\( T^{6} - \)\(18\!\cdots\!22\)\( p^{7} T^{7} + 6554166746730256755 p^{14} T^{8} - 129394076389776 p^{21} T^{9} + 196792867 p^{29} T^{10} - 46190 p^{35} T^{11} + p^{42} T^{12} \) | |
| 23 | \( 1 - 8714 T + 12235064070 T^{2} - 52572832791886 T^{3} + 75124527170447144431 T^{4} - \)\(92\!\cdots\!08\)\( T^{5} + \)\(30\!\cdots\!80\)\( T^{6} - \)\(92\!\cdots\!08\)\( p^{7} T^{7} + 75124527170447144431 p^{14} T^{8} - 52572832791886 p^{21} T^{9} + 12235064070 p^{28} T^{10} - 8714 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( 1 + 45222 T + 33208517399 T^{2} - 1997500194789810 T^{3} + \)\(42\!\cdots\!35\)\( T^{4} - \)\(62\!\cdots\!80\)\( T^{5} + \)\(83\!\cdots\!50\)\( T^{6} - \)\(62\!\cdots\!80\)\( p^{7} T^{7} + \)\(42\!\cdots\!35\)\( p^{14} T^{8} - 1997500194789810 p^{21} T^{9} + 33208517399 p^{28} T^{10} + 45222 p^{35} T^{11} + p^{42} T^{12} \) | |
| 31 | \( 1 - 264888 T + 96255834066 T^{2} - 13510523237414656 T^{3} + \)\(41\!\cdots\!67\)\( T^{4} - \)\(60\!\cdots\!36\)\( T^{5} + \)\(15\!\cdots\!92\)\( T^{6} - \)\(60\!\cdots\!36\)\( p^{7} T^{7} + \)\(41\!\cdots\!67\)\( p^{14} T^{8} - 13510523237414656 p^{21} T^{9} + 96255834066 p^{28} T^{10} - 264888 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( 1 - 496574 T + 472549781239 T^{2} - 182929318185061386 T^{3} + \)\(10\!\cdots\!43\)\( T^{4} - \)\(30\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!42\)\( T^{6} - \)\(30\!\cdots\!24\)\( p^{7} T^{7} + \)\(10\!\cdots\!43\)\( p^{14} T^{8} - 182929318185061386 p^{21} T^{9} + 472549781239 p^{28} T^{10} - 496574 p^{35} T^{11} + p^{42} T^{12} \) | |
| 41 | \( 1 - 128278 T + 754700218514 T^{2} - 65128194926371926 T^{3} + \)\(26\!\cdots\!39\)\( T^{4} - \)\(16\!\cdots\!72\)\( T^{5} + \)\(61\!\cdots\!44\)\( T^{6} - \)\(16\!\cdots\!72\)\( p^{7} T^{7} + \)\(26\!\cdots\!39\)\( p^{14} T^{8} - 65128194926371926 p^{21} T^{9} + 754700218514 p^{28} T^{10} - 128278 p^{35} T^{11} + p^{42} T^{12} \) | |
| 43 | \( 1 - 333730 T + 913489055742 T^{2} - 398035828093438414 T^{3} + \)\(42\!\cdots\!71\)\( T^{4} - \)\(20\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!92\)\( T^{6} - \)\(20\!\cdots\!40\)\( p^{7} T^{7} + \)\(42\!\cdots\!71\)\( p^{14} T^{8} - 398035828093438414 p^{21} T^{9} + 913489055742 p^{28} T^{10} - 333730 p^{35} T^{11} + p^{42} T^{12} \) | |
| 47 | \( 1 - 296282 T + 1030877085189 T^{2} - 237769985390371108 T^{3} + \)\(44\!\cdots\!15\)\( T^{4} - \)\(18\!\cdots\!42\)\( T^{5} + \)\(18\!\cdots\!02\)\( T^{6} - \)\(18\!\cdots\!42\)\( p^{7} T^{7} + \)\(44\!\cdots\!15\)\( p^{14} T^{8} - 237769985390371108 p^{21} T^{9} + 1030877085189 p^{28} T^{10} - 296282 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( 1 - 1629194 T + 6160842938282 T^{2} - 7967906071212665562 T^{3} + \)\(16\!\cdots\!19\)\( T^{4} - \)\(17\!\cdots\!40\)\( T^{5} + \)\(25\!\cdots\!96\)\( T^{6} - \)\(17\!\cdots\!40\)\( p^{7} T^{7} + \)\(16\!\cdots\!19\)\( p^{14} T^{8} - 7967906071212665562 p^{21} T^{9} + 6160842938282 p^{28} T^{10} - 1629194 p^{35} T^{11} + p^{42} T^{12} \) | |
| 59 | \( 1 - 1668736 T + 8805088487249 T^{2} - 3010131388108970220 T^{3} + \)\(16\!\cdots\!75\)\( T^{4} + \)\(34\!\cdots\!40\)\( T^{5} + \)\(21\!\cdots\!90\)\( T^{6} + \)\(34\!\cdots\!40\)\( p^{7} T^{7} + \)\(16\!\cdots\!75\)\( p^{14} T^{8} - 3010131388108970220 p^{21} T^{9} + 8805088487249 p^{28} T^{10} - 1668736 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 - 505900 T + 4699529815050 T^{2} - 6116559418988872492 T^{3} + \)\(18\!\cdots\!99\)\( T^{4} - \)\(22\!\cdots\!04\)\( T^{5} + \)\(87\!\cdots\!12\)\( T^{6} - \)\(22\!\cdots\!04\)\( p^{7} T^{7} + \)\(18\!\cdots\!99\)\( p^{14} T^{8} - 6116559418988872492 p^{21} T^{9} + 4699529815050 p^{28} T^{10} - 505900 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 - 5007524 T + 34911565698569 T^{2} - \)\(13\!\cdots\!76\)\( T^{3} + \)\(50\!\cdots\!23\)\( T^{4} - \)\(14\!\cdots\!44\)\( T^{5} + \)\(40\!\cdots\!02\)\( T^{6} - \)\(14\!\cdots\!44\)\( p^{7} T^{7} + \)\(50\!\cdots\!23\)\( p^{14} T^{8} - \)\(13\!\cdots\!76\)\( p^{21} T^{9} + 34911565698569 p^{28} T^{10} - 5007524 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( 1 - 5049052 T + 46208905840970 T^{2} - \)\(18\!\cdots\!36\)\( T^{3} + \)\(97\!\cdots\!27\)\( T^{4} - \)\(30\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} - \)\(30\!\cdots\!40\)\( p^{7} T^{7} + \)\(97\!\cdots\!27\)\( p^{14} T^{8} - \)\(18\!\cdots\!36\)\( p^{21} T^{9} + 46208905840970 p^{28} T^{10} - 5049052 p^{35} T^{11} + p^{42} T^{12} \) | |
| 73 | \( 1 - 4623188 T + 63776883503779 T^{2} - \)\(23\!\cdots\!86\)\( T^{3} + \)\(17\!\cdots\!27\)\( T^{4} - \)\(49\!\cdots\!34\)\( T^{5} + \)\(25\!\cdots\!22\)\( T^{6} - \)\(49\!\cdots\!34\)\( p^{7} T^{7} + \)\(17\!\cdots\!27\)\( p^{14} T^{8} - \)\(23\!\cdots\!86\)\( p^{21} T^{9} + 63776883503779 p^{28} T^{10} - 4623188 p^{35} T^{11} + p^{42} T^{12} \) | |
| 79 | \( 1 - 2266540 T + 36998705935114 T^{2} - 48446079615533389412 T^{3} + \)\(11\!\cdots\!15\)\( T^{4} - \)\(10\!\cdots\!24\)\( T^{5} + \)\(21\!\cdots\!60\)\( T^{6} - \)\(10\!\cdots\!24\)\( p^{7} T^{7} + \)\(11\!\cdots\!15\)\( p^{14} T^{8} - 48446079615533389412 p^{21} T^{9} + 36998705935114 p^{28} T^{10} - 2266540 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( 1 + 492020 T + 86803547616290 T^{2} + 4886887226584325796 T^{3} + \)\(40\!\cdots\!19\)\( T^{4} - \)\(79\!\cdots\!68\)\( T^{5} + \)\(12\!\cdots\!40\)\( T^{6} - \)\(79\!\cdots\!68\)\( p^{7} T^{7} + \)\(40\!\cdots\!19\)\( p^{14} T^{8} + 4886887226584325796 p^{21} T^{9} + 86803547616290 p^{28} T^{10} + 492020 p^{35} T^{11} + p^{42} T^{12} \) | |
| 89 | \( 1 - 10355792 T + 182312193693834 T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!67\)\( T^{4} - \)\(60\!\cdots\!60\)\( T^{5} + \)\(62\!\cdots\!76\)\( T^{6} - \)\(60\!\cdots\!60\)\( p^{7} T^{7} + \)\(13\!\cdots\!67\)\( p^{14} T^{8} - \)\(11\!\cdots\!00\)\( p^{21} T^{9} + 182312193693834 p^{28} T^{10} - 10355792 p^{35} T^{11} + p^{42} T^{12} \) | |
| 97 | \( 1 - 7245546 T + 273016178957018 T^{2} - \)\(56\!\cdots\!46\)\( T^{3} + \)\(23\!\cdots\!19\)\( T^{4} + \)\(88\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!64\)\( T^{6} + \)\(88\!\cdots\!08\)\( p^{7} T^{7} + \)\(23\!\cdots\!19\)\( p^{14} T^{8} - \)\(56\!\cdots\!46\)\( p^{21} T^{9} + 273016178957018 p^{28} T^{10} - 7245546 p^{35} T^{11} + p^{42} T^{12} \) | |
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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
−4.85089674033245333237355327104, −4.23745596572228991605910603354, −4.19375407466891455565146348997, −4.15721035789479749499640667100, −4.07449717355913532357855482822, −3.90645507079347419363984611782, −3.78813169943982576790180194400, −3.59932130694629170352522587255, −3.24138456578442375535698509111, −3.14145638121285590180944506103, −3.12274358075301841540311640320, −3.02947767869505372686925983376, −2.78261040381991680147206073194, −2.15707643676194171241570054561, −2.13057197446442330822685615937, −2.09867770879390009004641145672, −2.05246213509956314593833031848, −2.04485095295584372368533860957, −1.86199982705456182197499044819, −1.20962461036006136475792324760, −1.18717849948479674727300881804, −1.08157207138869149288828330453, −0.942921188681978834024473211852, −0.809390950982020086474540622515, −0.74751430237423425557077997988, 0.74751430237423425557077997988, 0.809390950982020086474540622515, 0.942921188681978834024473211852, 1.08157207138869149288828330453, 1.18717849948479674727300881804, 1.20962461036006136475792324760, 1.86199982705456182197499044819, 2.04485095295584372368533860957, 2.05246213509956314593833031848, 2.09867770879390009004641145672, 2.13057197446442330822685615937, 2.15707643676194171241570054561, 2.78261040381991680147206073194, 3.02947767869505372686925983376, 3.12274358075301841540311640320, 3.14145638121285590180944506103, 3.24138456578442375535698509111, 3.59932130694629170352522587255, 3.78813169943982576790180194400, 3.90645507079347419363984611782, 4.07449717355913532357855482822, 4.15721035789479749499640667100, 4.19375407466891455565146348997, 4.23745596572228991605910603354, 4.85089674033245333237355327104
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.