Dirichlet series
| L(s) = 1 | − 24·2-s + 192·4-s − 110·5-s + 635·7-s + 1.02e3·8-s + 2.64e3·10-s + 548·11-s + 1.98e4·13-s − 1.52e4·14-s − 3.68e4·16-s + 2.09e4·17-s + 2.83e4·19-s − 2.11e4·20-s − 1.31e4·22-s + 3.27e4·23-s + 1.34e5·25-s − 4.77e5·26-s + 1.21e5·28-s − 2.34e5·29-s − 1.47e5·31-s + 2.94e5·32-s − 5.03e5·34-s − 6.98e4·35-s − 3.67e5·37-s − 6.81e5·38-s − 1.12e5·40-s + 2.87e6·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s − 0.393·5-s + 0.699·7-s + 0.707·8-s + 0.834·10-s + 0.124·11-s + 2.51·13-s − 1.48·14-s − 9/4·16-s + 1.03·17-s + 0.949·19-s − 0.590·20-s − 0.263·22-s + 0.560·23-s + 1.72·25-s − 5.32·26-s + 1.04·28-s − 1.78·29-s − 0.891·31-s + 1.59·32-s − 2.19·34-s − 0.275·35-s − 1.19·37-s − 2.01·38-s − 0.278·40-s + 6.51·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{12} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.71847\times 10^{9}\) |
| Root analytic conductor: | \(6.27379\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(6.650618336\) |
| \(L(\frac12)\) | \(\approx\) | \(6.650618336\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{3} T + p^{6} T^{2} )^{3} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 635 T - 72748 p T^{2} + 667945 p^{4} T^{3} - 72748 p^{8} T^{4} - 635 p^{14} T^{5} + p^{21} T^{6} \) | |
| good | 5 | \( 1 + 22 p T - 24502 p T^{2} - 6966776 p T^{3} + 194927678 p^{2} T^{4} + 67954540106 p^{2} T^{5} + 3210602234854 p^{2} T^{6} + 67954540106 p^{9} T^{7} + 194927678 p^{16} T^{8} - 6966776 p^{22} T^{9} - 24502 p^{29} T^{10} + 22 p^{36} T^{11} + p^{42} T^{12} \) |
| 11 | \( 1 - 548 T - 4276622 T^{2} - 160652942204 T^{3} - 24911416435726 T^{4} + 22869026595445016 p T^{5} + \)\(16\!\cdots\!26\)\( p^{2} T^{6} + 22869026595445016 p^{8} T^{7} - 24911416435726 p^{14} T^{8} - 160652942204 p^{21} T^{9} - 4276622 p^{28} T^{10} - 548 p^{35} T^{11} + p^{42} T^{12} \) | |
| 13 | \( ( 1 - 9949 T + 49691323 T^{2} + 136109955742 T^{3} + 49691323 p^{7} T^{4} - 9949 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 17 | \( 1 - 20972 T - 705540803 T^{2} + 6376043535148 T^{3} + 523135254353192486 T^{4} - \)\(12\!\cdots\!16\)\( T^{5} - \)\(24\!\cdots\!03\)\( T^{6} - \)\(12\!\cdots\!16\)\( p^{7} T^{7} + 523135254353192486 p^{14} T^{8} + 6376043535148 p^{21} T^{9} - 705540803 p^{28} T^{10} - 20972 p^{35} T^{11} + p^{42} T^{12} \) | |
| 19 | \( 1 - 28383 T - 1114469436 T^{2} + 54482095066865 T^{3} + 423775802745621540 T^{4} - \)\(15\!\cdots\!97\)\( p T^{5} + \)\(12\!\cdots\!54\)\( p^{2} T^{6} - \)\(15\!\cdots\!97\)\( p^{8} T^{7} + 423775802745621540 p^{14} T^{8} + 54482095066865 p^{21} T^{9} - 1114469436 p^{28} T^{10} - 28383 p^{35} T^{11} + p^{42} T^{12} \) | |
| 23 | \( 1 - 32732 T - 4257974405 T^{2} + 384004545798916 T^{3} + 1117012122760959974 T^{4} - \)\(67\!\cdots\!44\)\( T^{5} + \)\(47\!\cdots\!87\)\( T^{6} - \)\(67\!\cdots\!44\)\( p^{7} T^{7} + 1117012122760959974 p^{14} T^{8} + 384004545798916 p^{21} T^{9} - 4257974405 p^{28} T^{10} - 32732 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( ( 1 + 117088 T + 32964581490 T^{2} + 4122277361969500 T^{3} + 32964581490 p^{7} T^{4} + 117088 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 31 | \( 1 + 147865 T + 804539397 T^{2} + 5946198810653250 T^{3} + 73494989034467037151 T^{4} - \)\(89\!\cdots\!95\)\( T^{5} + \)\(14\!\cdots\!38\)\( T^{6} - \)\(89\!\cdots\!95\)\( p^{7} T^{7} + 73494989034467037151 p^{14} T^{8} + 5946198810653250 p^{21} T^{9} + 804539397 p^{28} T^{10} + 147865 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( 1 + 367503 T - 25474888698 T^{2} + 9103014313145033 T^{3} + \)\(13\!\cdots\!36\)\( T^{4} - \)\(38\!\cdots\!61\)\( T^{5} - \)\(13\!\cdots\!08\)\( T^{6} - \)\(38\!\cdots\!61\)\( p^{7} T^{7} + \)\(13\!\cdots\!36\)\( p^{14} T^{8} + 9103014313145033 p^{21} T^{9} - 25474888698 p^{28} T^{10} + 367503 p^{35} T^{11} + p^{42} T^{12} \) | |
| 41 | \( ( 1 - 1437954 T + 1203779247675 T^{2} - 639058142642674020 T^{3} + 1203779247675 p^{7} T^{4} - 1437954 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 43 | \( ( 1 - 749397 T + 509340737649 T^{2} - 170374599838662482 T^{3} + 509340737649 p^{7} T^{4} - 749397 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 47 | \( 1 + 741486 T - 864917943045 T^{2} - 363526634183852922 T^{3} + \)\(76\!\cdots\!34\)\( T^{4} + \)\(12\!\cdots\!42\)\( T^{5} - \)\(39\!\cdots\!57\)\( T^{6} + \)\(12\!\cdots\!42\)\( p^{7} T^{7} + \)\(76\!\cdots\!34\)\( p^{14} T^{8} - 363526634183852922 p^{21} T^{9} - 864917943045 p^{28} T^{10} + 741486 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( 1 + 1032432 T - 2372684155170 T^{2} - 1282239454997266536 T^{3} + \)\(52\!\cdots\!54\)\( T^{4} + \)\(13\!\cdots\!84\)\( T^{5} - \)\(62\!\cdots\!98\)\( T^{6} + \)\(13\!\cdots\!84\)\( p^{7} T^{7} + \)\(52\!\cdots\!54\)\( p^{14} T^{8} - 1282239454997266536 p^{21} T^{9} - 2372684155170 p^{28} T^{10} + 1032432 p^{35} T^{11} + p^{42} T^{12} \) | |
| 59 | \( 1 + 2389238 T - 54827765486 T^{2} - 423685096884345220 T^{3} + \)\(19\!\cdots\!90\)\( T^{4} - \)\(12\!\cdots\!02\)\( T^{5} - \)\(36\!\cdots\!66\)\( T^{6} - \)\(12\!\cdots\!02\)\( p^{7} T^{7} + \)\(19\!\cdots\!90\)\( p^{14} T^{8} - 423685096884345220 p^{21} T^{9} - 54827765486 p^{28} T^{10} + 2389238 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 + 1238746 T - 3495698274519 T^{2} - 5946787860873446010 T^{3} + \)\(19\!\cdots\!30\)\( T^{4} + \)\(36\!\cdots\!66\)\( T^{5} + \)\(44\!\cdots\!41\)\( T^{6} + \)\(36\!\cdots\!66\)\( p^{7} T^{7} + \)\(19\!\cdots\!30\)\( p^{14} T^{8} - 5946787860873446010 p^{21} T^{9} - 3495698274519 p^{28} T^{10} + 1238746 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 - 2462497 T - 13268883047308 T^{2} + 11883615447576942903 T^{3} + \)\(18\!\cdots\!76\)\( T^{4} - \)\(88\!\cdots\!01\)\( T^{5} - \)\(11\!\cdots\!98\)\( T^{6} - \)\(88\!\cdots\!01\)\( p^{7} T^{7} + \)\(18\!\cdots\!76\)\( p^{14} T^{8} + 11883615447576942903 p^{21} T^{9} - 13268883047308 p^{28} T^{10} - 2462497 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( ( 1 - 636262 T + 18637597322721 T^{2} - 19723903310040373948 T^{3} + 18637597322721 p^{7} T^{4} - 636262 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 73 | \( 1 + 4609961 T - 504249505402 T^{2} + 439056397955388279 T^{3} + \)\(28\!\cdots\!16\)\( T^{4} - \)\(56\!\cdots\!07\)\( T^{5} - \)\(26\!\cdots\!12\)\( T^{6} - \)\(56\!\cdots\!07\)\( p^{7} T^{7} + \)\(28\!\cdots\!16\)\( p^{14} T^{8} + 439056397955388279 p^{21} T^{9} - 504249505402 p^{28} T^{10} + 4609961 p^{35} T^{11} + p^{42} T^{12} \) | |
| 79 | \( 1 - 6152849 T - 9996162298563 T^{2} + 91000866673663082982 T^{3} + \)\(18\!\cdots\!79\)\( T^{4} - \)\(18\!\cdots\!97\)\( T^{5} - \)\(66\!\cdots\!06\)\( T^{6} - \)\(18\!\cdots\!97\)\( p^{7} T^{7} + \)\(18\!\cdots\!79\)\( p^{14} T^{8} + 91000866673663082982 p^{21} T^{9} - 9996162298563 p^{28} T^{10} - 6152849 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( ( 1 - 12029884 T + 1146704654226 p T^{2} - \)\(54\!\cdots\!38\)\( T^{3} + 1146704654226 p^{8} T^{4} - 12029884 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 89 | \( 1 - 10646976 T - 54532854429783 T^{2} + \)\(16\!\cdots\!68\)\( T^{3} + \)\(11\!\cdots\!46\)\( p T^{4} - \)\(26\!\cdots\!28\)\( T^{5} - \)\(34\!\cdots\!11\)\( T^{6} - \)\(26\!\cdots\!28\)\( p^{7} T^{7} + \)\(11\!\cdots\!46\)\( p^{15} T^{8} + \)\(16\!\cdots\!68\)\( p^{21} T^{9} - 54532854429783 p^{28} T^{10} - 10646976 p^{35} T^{11} + p^{42} T^{12} \) | |
| 97 | \( ( 1 - 2281472 T + 178988504234792 T^{2} - \)\(22\!\cdots\!46\)\( T^{3} + 178988504234792 p^{7} T^{4} - 2281472 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
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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
−6.01973626278208092471213158021, −5.83166757748642801540207606544, −5.58211952386985561247781427824, −5.57218265079979756354541860092, −5.22935485224327541944287031491, −5.05739046439857781105933604082, −4.62631315490536333618286752295, −4.31238868873235493760234174345, −4.27723869605522784613479980349, −4.20767994870684937214676651382, −3.70247488275758409797491554616, −3.52893136519863071412030627101, −3.42456934179565188840308524022, −3.16796976258846330522294123548, −2.53960548936687602932248939393, −2.49664301078676126355645984811, −2.44514059241400705087522557982, −1.66757975580888226144647376085, −1.56925752569469154549875158329, −1.44712759021640165564308646118, −1.17353125681665586864323362846, −0.74733985993735758465978392881, −0.66595292770718486017135440211, −0.66449043974912199091388707117, −0.39601269354382914309386226796, 0.39601269354382914309386226796, 0.66449043974912199091388707117, 0.66595292770718486017135440211, 0.74733985993735758465978392881, 1.17353125681665586864323362846, 1.44712759021640165564308646118, 1.56925752569469154549875158329, 1.66757975580888226144647376085, 2.44514059241400705087522557982, 2.49664301078676126355645984811, 2.53960548936687602932248939393, 3.16796976258846330522294123548, 3.42456934179565188840308524022, 3.52893136519863071412030627101, 3.70247488275758409797491554616, 4.20767994870684937214676651382, 4.27723869605522784613479980349, 4.31238868873235493760234174345, 4.62631315490536333618286752295, 5.05739046439857781105933604082, 5.22935485224327541944287031491, 5.57218265079979756354541860092, 5.58211952386985561247781427824, 5.83166757748642801540207606544, 6.01973626278208092471213158021
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.