Dirichlet series
| L(s) = 1 | − 32·3-s + 264·7-s + 512·9-s + 2.20e3·11-s − 858·13-s + 3.27e3·17-s − 8.44e3·21-s − 1.99e4·23-s + 2.52e4·27-s + 1.04e5·31-s − 7.04e4·33-s + 2.41e5·37-s + 2.74e4·39-s + 3.51e5·41-s − 6.07e4·43-s + 3.55e5·47-s + 3.48e4·49-s − 1.04e5·51-s − 3.46e5·53-s − 4.92e5·61-s + 1.35e5·63-s + 2.30e5·67-s + 6.39e5·69-s + 1.74e5·71-s + 3.32e5·73-s + 5.80e5·77-s − 1.37e6·81-s + ⋯ |
| L(s) = 1 | − 1.18·3-s + 0.769·7-s + 0.702·9-s + 1.65·11-s − 0.390·13-s + 0.667·17-s − 0.912·21-s − 1.64·23-s + 1.28·27-s + 3.52·31-s − 1.95·33-s + 4.76·37-s + 0.462·39-s + 5.10·41-s − 0.763·43-s + 3.42·47-s + 0.296·49-s − 0.790·51-s − 2.32·53-s − 2.17·61-s + 0.540·63-s + 0.765·67-s + 1.94·69-s + 0.486·71-s + 0.854·73-s + 1.27·77-s − 2.58·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+3)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{12} \cdot 5^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.48244\times 10^{8}\) |
| Root analytic conductor: | \(4.79639\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{12} \cdot 5^{12} ,\ ( \ : [3]^{6} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(18.14041154\) |
| \(L(\frac12)\) | \(\approx\) | \(18.14041154\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) | |
| good | 3 | \( 1 + 32 T + 512 T^{2} - 104 p^{5} T^{3} - 56345 p^{2} T^{4} + 1086392 p^{2} T^{5} + 99095072 p^{2} T^{6} + 1086392 p^{8} T^{7} - 56345 p^{14} T^{8} - 104 p^{23} T^{9} + 512 p^{24} T^{10} + 32 p^{30} T^{11} + p^{36} T^{12} \) |
| 7 | \( 1 - 264 T + 34848 T^{2} + 48340864 T^{3} - 13585926801 T^{4} - 481340689896 p T^{5} + 51660806902496 p^{2} T^{6} - 481340689896 p^{7} T^{7} - 13585926801 p^{12} T^{8} + 48340864 p^{18} T^{9} + 34848 p^{24} T^{10} - 264 p^{30} T^{11} + p^{36} T^{12} \) | |
| 11 | \( ( 1 - 100 p T + 276253 p T^{2} - 20190200 p^{2} T^{3} + 276253 p^{7} T^{4} - 100 p^{13} T^{5} + p^{18} T^{6} )^{2} \) | |
| 13 | \( 1 + 66 p T + 2178 p^{2} T^{2} + 2806723722 T^{3} + 30691727023695 T^{4} + 22592900379705612 T^{5} + 12026485285270478748 T^{6} + 22592900379705612 p^{6} T^{7} + 30691727023695 p^{12} T^{8} + 2806723722 p^{18} T^{9} + 2178 p^{26} T^{10} + 66 p^{31} T^{11} + p^{36} T^{12} \) | |
| 17 | \( 1 - 3278 T + 5372642 T^{2} - 644448046 p T^{3} - 1428673989025 p^{2} T^{4} + 148541287751516 p^{3} T^{5} - 1364018275090372 p^{4} T^{6} + 148541287751516 p^{9} T^{7} - 1428673989025 p^{14} T^{8} - 644448046 p^{19} T^{9} + 5372642 p^{24} T^{10} - 3278 p^{30} T^{11} + p^{36} T^{12} \) | |
| 19 | \( 1 - 167166534 T^{2} + 13715242279167135 T^{4} - \)\(74\!\cdots\!00\)\( T^{6} + 13715242279167135 p^{12} T^{8} - 167166534 p^{24} T^{10} + p^{36} T^{12} \) | |
| 23 | \( 1 + 19984 T + 199680128 T^{2} + 3629111359576 T^{3} + 87866945710300079 T^{4} + \)\(99\!\cdots\!72\)\( T^{5} + \)\(89\!\cdots\!24\)\( T^{6} + \)\(99\!\cdots\!72\)\( p^{6} T^{7} + 87866945710300079 p^{12} T^{8} + 3629111359576 p^{18} T^{9} + 199680128 p^{24} T^{10} + 19984 p^{30} T^{11} + p^{36} T^{12} \) | |
| 29 | \( 1 - 1033291734 T^{2} + 1369068699154031775 T^{4} - \)\(75\!\cdots\!40\)\( T^{6} + 1369068699154031775 p^{12} T^{8} - 1033291734 p^{24} T^{10} + p^{36} T^{12} \) | |
| 31 | \( ( 1 - 52488 T + 2507733591 T^{2} - 91420641186592 T^{3} + 2507733591 p^{6} T^{4} - 52488 p^{12} T^{5} + p^{18} T^{6} )^{2} \) | |
| 37 | \( 1 - 241554 T + 29174167458 T^{2} - 2703692719264386 T^{3} + \)\(21\!\cdots\!19\)\( T^{4} - \)\(13\!\cdots\!52\)\( T^{5} + \)\(73\!\cdots\!04\)\( T^{6} - \)\(13\!\cdots\!52\)\( p^{6} T^{7} + \)\(21\!\cdots\!19\)\( p^{12} T^{8} - 2703692719264386 p^{18} T^{9} + 29174167458 p^{24} T^{10} - 241554 p^{30} T^{11} + p^{36} T^{12} \) | |
| 41 | \( ( 1 - 175868 T + 20110360631 T^{2} - 1685864350808792 T^{3} + 20110360631 p^{6} T^{4} - 175868 p^{12} T^{5} + p^{18} T^{6} )^{2} \) | |
| 43 | \( 1 + 60720 T + 1843459200 T^{2} + 170170651040280 T^{3} - 39718848676435556097 T^{4} - \)\(33\!\cdots\!60\)\( T^{5} - \)\(11\!\cdots\!00\)\( T^{6} - \)\(33\!\cdots\!60\)\( p^{6} T^{7} - 39718848676435556097 p^{12} T^{8} + 170170651040280 p^{18} T^{9} + 1843459200 p^{24} T^{10} + 60720 p^{30} T^{11} + p^{36} T^{12} \) | |
| 47 | \( 1 - 355248 T + 63100570752 T^{2} - 8734381777271592 T^{3} + \)\(12\!\cdots\!95\)\( T^{4} - \)\(15\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!08\)\( T^{6} - \)\(15\!\cdots\!92\)\( p^{6} T^{7} + \)\(12\!\cdots\!95\)\( p^{12} T^{8} - 8734381777271592 p^{18} T^{9} + 63100570752 p^{24} T^{10} - 355248 p^{30} T^{11} + p^{36} T^{12} \) | |
| 53 | \( 1 + 346526 T + 60040134338 T^{2} + 13491114635468654 T^{3} + \)\(34\!\cdots\!59\)\( T^{4} + \)\(52\!\cdots\!68\)\( T^{5} + \)\(67\!\cdots\!84\)\( T^{6} + \)\(52\!\cdots\!68\)\( p^{6} T^{7} + \)\(34\!\cdots\!59\)\( p^{12} T^{8} + 13491114635468654 p^{18} T^{9} + 60040134338 p^{24} T^{10} + 346526 p^{30} T^{11} + p^{36} T^{12} \) | |
| 59 | \( 1 - 133435411878 T^{2} + \)\(95\!\cdots\!71\)\( T^{4} - \)\(47\!\cdots\!12\)\( T^{6} + \)\(95\!\cdots\!71\)\( p^{12} T^{8} - 133435411878 p^{24} T^{10} + p^{36} T^{12} \) | |
| 61 | \( ( 1 + 246444 T + 82272859695 T^{2} + 7520154028705560 T^{3} + 82272859695 p^{6} T^{4} + 246444 p^{12} T^{5} + p^{18} T^{6} )^{2} \) | |
| 67 | \( 1 - 230304 T + 26519966208 T^{2} + 6500435613362824 T^{3} - \)\(29\!\cdots\!61\)\( T^{4} - \)\(19\!\cdots\!12\)\( T^{5} + \)\(53\!\cdots\!24\)\( T^{6} - \)\(19\!\cdots\!12\)\( p^{6} T^{7} - \)\(29\!\cdots\!61\)\( p^{12} T^{8} + 6500435613362824 p^{18} T^{9} + 26519966208 p^{24} T^{10} - 230304 p^{30} T^{11} + p^{36} T^{12} \) | |
| 71 | \( ( 1 - 87064 T + 266865978695 T^{2} - 4191185495253760 T^{3} + 266865978695 p^{6} T^{4} - 87064 p^{12} T^{5} + p^{18} T^{6} )^{2} \) | |
| 73 | \( 1 - 4554 p T + 10369458 p^{2} T^{2} + 3861289507981862 T^{3} - \)\(18\!\cdots\!25\)\( T^{4} + \)\(17\!\cdots\!32\)\( T^{5} + \)\(42\!\cdots\!28\)\( T^{6} + \)\(17\!\cdots\!32\)\( p^{6} T^{7} - \)\(18\!\cdots\!25\)\( p^{12} T^{8} + 3861289507981862 p^{18} T^{9} + 10369458 p^{26} T^{10} - 4554 p^{31} T^{11} + p^{36} T^{12} \) | |
| 79 | \( 1 - 3039466842 p T^{2} + \)\(85\!\cdots\!31\)\( T^{4} - \)\(34\!\cdots\!92\)\( T^{6} + \)\(85\!\cdots\!31\)\( p^{12} T^{8} - 3039466842 p^{25} T^{10} + p^{36} T^{12} \) | |
| 83 | \( 1 - 2190936 T + 2400100278048 T^{2} - 2073409539706878384 T^{3} + \)\(15\!\cdots\!39\)\( T^{4} - \)\(10\!\cdots\!08\)\( T^{5} + \)\(60\!\cdots\!44\)\( T^{6} - \)\(10\!\cdots\!08\)\( p^{6} T^{7} + \)\(15\!\cdots\!39\)\( p^{12} T^{8} - 2073409539706878384 p^{18} T^{9} + 2400100278048 p^{24} T^{10} - 2190936 p^{30} T^{11} + p^{36} T^{12} \) | |
| 89 | \( 1 - 2595301222374 T^{2} + \)\(29\!\cdots\!55\)\( T^{4} - \)\(19\!\cdots\!20\)\( T^{6} + \)\(29\!\cdots\!55\)\( p^{12} T^{8} - 2595301222374 p^{24} T^{10} + p^{36} T^{12} \) | |
| 97 | \( 1 + 3338406 T + 5572477310418 T^{2} + 7517418539616763974 T^{3} + \)\(94\!\cdots\!59\)\( T^{4} + \)\(10\!\cdots\!08\)\( T^{5} + \)\(96\!\cdots\!24\)\( T^{6} + \)\(10\!\cdots\!08\)\( p^{6} T^{7} + \)\(94\!\cdots\!59\)\( p^{12} T^{8} + 7517418539616763974 p^{18} T^{9} + 5572477310418 p^{24} T^{10} + 3338406 p^{30} T^{11} + p^{36} T^{12} \) | |
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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
−6.50484809662911815782597094106, −6.07634660342150257632582340673, −6.07274279153647240034094073600, −5.89488819000115257611604714083, −5.84641152171666468505960922822, −5.70950739903893929092449563371, −5.16001421663580463306447804646, −4.73219411413589945213877672539, −4.60611946647183137635140889697, −4.51701845992357806987059447674, −4.38035685085508855923891359055, −4.17060289839426024891925014911, −3.96402803333423978188953614015, −3.60045537976856658277873939944, −3.08848410951634583632341006427, −2.68707920856107799560570292569, −2.66143473067366629814445472002, −2.59559729006278690387949629786, −1.97046101156361156850597703227, −1.74072381250167036873202386908, −1.29515748775667959414545148721, −0.809746636791909231822714875171, −0.75293351809265766978540252538, −0.72388957000807713316531203984, −0.61657793566922504885119076203, 0.61657793566922504885119076203, 0.72388957000807713316531203984, 0.75293351809265766978540252538, 0.809746636791909231822714875171, 1.29515748775667959414545148721, 1.74072381250167036873202386908, 1.97046101156361156850597703227, 2.59559729006278690387949629786, 2.66143473067366629814445472002, 2.68707920856107799560570292569, 3.08848410951634583632341006427, 3.60045537976856658277873939944, 3.96402803333423978188953614015, 4.17060289839426024891925014911, 4.38035685085508855923891359055, 4.51701845992357806987059447674, 4.60611946647183137635140889697, 4.73219411413589945213877672539, 5.16001421663580463306447804646, 5.70950739903893929092449563371, 5.84641152171666468505960922822, 5.89488819000115257611604714083, 6.07274279153647240034094073600, 6.07634660342150257632582340673, 6.50484809662911815782597094106
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.