Dirichlet series
| L(s) = 1 | + 38·3-s − 29·4-s − 150·5-s + 220·7-s + 8·8-s + 252·9-s − 170·11-s − 1.10e3·12-s + 1.01e3·13-s − 5.70e3·15-s + 225·16-s + 728·17-s + 1.21e3·19-s + 4.35e3·20-s + 8.36e3·21-s + 8.95e3·23-s + 304·24-s + 1.31e4·25-s − 7.42e3·27-s − 6.38e3·28-s + 8.36e3·29-s + 2.86e3·31-s − 6.32e3·32-s − 6.46e3·33-s − 3.30e4·35-s − 7.30e3·36-s + 1.38e4·37-s + ⋯ |
| L(s) = 1 | + 2.43·3-s − 0.906·4-s − 2.68·5-s + 1.69·7-s + 0.0441·8-s + 1.03·9-s − 0.423·11-s − 2.20·12-s + 1.66·13-s − 6.54·15-s + 0.219·16-s + 0.610·17-s + 0.774·19-s + 2.43·20-s + 4.13·21-s + 3.52·23-s + 0.107·24-s + 21/5·25-s − 1.95·27-s − 1.53·28-s + 1.84·29-s + 0.534·31-s − 1.09·32-s − 1.03·33-s − 4.55·35-s − 0.939·36-s + 1.66·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(5^{6} \cdot 13^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.28364\times 10^{6}\) |
| Root analytic conductor: | \(3.22876\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 5^{6} \cdot 13^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(12.31059372\) |
| \(L(\frac12)\) | \(\approx\) | \(12.31059372\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + p^{2} T )^{6} \) |
| 13 | \( ( 1 - p^{2} T )^{6} \) | |
| good | 2 | \( 1 + 29 T^{2} - p^{3} T^{3} + 77 p^{3} T^{4} + 183 p^{5} T^{5} + 1633 p^{4} T^{6} + 183 p^{10} T^{7} + 77 p^{13} T^{8} - p^{18} T^{9} + 29 p^{20} T^{10} + p^{30} T^{12} \) |
| 3 | \( 1 - 38 T + 1192 T^{2} - 28298 T^{3} + 588259 T^{4} - 10474916 T^{5} + 58129552 p T^{6} - 10474916 p^{5} T^{7} + 588259 p^{10} T^{8} - 28298 p^{15} T^{9} + 1192 p^{20} T^{10} - 38 p^{25} T^{11} + p^{30} T^{12} \) | |
| 7 | \( 1 - 220 T + 31670 T^{2} - 3730484 T^{3} + 87423337 p T^{4} - 114244097528 T^{5} + 20006370609140 T^{6} - 114244097528 p^{5} T^{7} + 87423337 p^{11} T^{8} - 3730484 p^{15} T^{9} + 31670 p^{20} T^{10} - 220 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 + 170 T + 389192 T^{2} + 69834742 T^{3} + 121500995251 T^{4} + 18037597885052 T^{5} + 21552475206990064 T^{6} + 18037597885052 p^{5} T^{7} + 121500995251 p^{10} T^{8} + 69834742 p^{15} T^{9} + 389192 p^{20} T^{10} + 170 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( 1 - 728 T + 7446418 T^{2} - 4570222968 T^{3} + 24561153797711 T^{4} - 12247773888449840 T^{5} + 45387161516448643708 T^{6} - 12247773888449840 p^{5} T^{7} + 24561153797711 p^{10} T^{8} - 4570222968 p^{15} T^{9} + 7446418 p^{20} T^{10} - 728 p^{25} T^{11} + p^{30} T^{12} \) | |
| 19 | \( 1 - 1218 T + 11846024 T^{2} - 10423704430 T^{3} + 60706159573955 T^{4} - 39930377364402508 T^{5} + \)\(18\!\cdots\!60\)\( T^{6} - 39930377364402508 p^{5} T^{7} + 60706159573955 p^{10} T^{8} - 10423704430 p^{15} T^{9} + 11846024 p^{20} T^{10} - 1218 p^{25} T^{11} + p^{30} T^{12} \) | |
| 23 | \( 1 - 8954 T + 56775080 T^{2} - 249513985342 T^{3} + 953168537700283 T^{4} - 2980267913654582300 T^{5} + \)\(83\!\cdots\!28\)\( T^{6} - 2980267913654582300 p^{5} T^{7} + 953168537700283 p^{10} T^{8} - 249513985342 p^{15} T^{9} + 56775080 p^{20} T^{10} - 8954 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 - 8364 T + 84596070 T^{2} - 566047110828 T^{3} + 3637365090566343 T^{4} - 19068610713544900344 T^{5} + \)\(95\!\cdots\!12\)\( T^{6} - 19068610713544900344 p^{5} T^{7} + 3637365090566343 p^{10} T^{8} - 566047110828 p^{15} T^{9} + 84596070 p^{20} T^{10} - 8364 p^{25} T^{11} + p^{30} T^{12} \) | |
| 31 | \( 1 - 2862 T + 127984424 T^{2} - 350113276842 T^{3} + 252141876570389 p T^{4} - 18638936625359359444 T^{5} + \)\(28\!\cdots\!08\)\( T^{6} - 18638936625359359444 p^{5} T^{7} + 252141876570389 p^{11} T^{8} - 350113276842 p^{15} T^{9} + 127984424 p^{20} T^{10} - 2862 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( 1 - 13840 T + 394436526 T^{2} - 3940885729680 T^{3} + 64424395368361751 T^{4} - \)\(49\!\cdots\!60\)\( T^{5} + \)\(58\!\cdots\!28\)\( T^{6} - \)\(49\!\cdots\!60\)\( p^{5} T^{7} + 64424395368361751 p^{10} T^{8} - 3940885729680 p^{15} T^{9} + 394436526 p^{20} T^{10} - 13840 p^{25} T^{11} + p^{30} T^{12} \) | |
| 41 | \( 1 - 2248 T + 101584354 T^{2} - 363275006568 T^{3} + 8660415768725663 T^{4} + 15192477336105119984 T^{5} + \)\(22\!\cdots\!08\)\( T^{6} + 15192477336105119984 p^{5} T^{7} + 8660415768725663 p^{10} T^{8} - 363275006568 p^{15} T^{9} + 101584354 p^{20} T^{10} - 2248 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 - 3822 T + 596663336 T^{2} - 1584621297202 T^{3} + 169737452440702643 T^{4} - \)\(31\!\cdots\!96\)\( T^{5} + \)\(30\!\cdots\!80\)\( T^{6} - \)\(31\!\cdots\!96\)\( p^{5} T^{7} + 169737452440702643 p^{10} T^{8} - 1584621297202 p^{15} T^{9} + 596663336 p^{20} T^{10} - 3822 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 + 10860 T + 795300422 T^{2} + 9421501134148 T^{3} + 334916282659819951 T^{4} + \)\(37\!\cdots\!52\)\( T^{5} + \)\(91\!\cdots\!64\)\( T^{6} + \)\(37\!\cdots\!52\)\( p^{5} T^{7} + 334916282659819951 p^{10} T^{8} + 9421501134148 p^{15} T^{9} + 795300422 p^{20} T^{10} + 10860 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( 1 - 6584 T + 1933839338 T^{2} - 9886956487864 T^{3} + 1731344925483066967 T^{4} - \)\(72\!\cdots\!28\)\( T^{5} + \)\(91\!\cdots\!44\)\( T^{6} - \)\(72\!\cdots\!28\)\( p^{5} T^{7} + 1731344925483066967 p^{10} T^{8} - 9886956487864 p^{15} T^{9} + 1933839338 p^{20} T^{10} - 6584 p^{25} T^{11} + p^{30} T^{12} \) | |
| 59 | \( 1 + 28874 T + 3221213064 T^{2} + 97504017632694 T^{3} + 4621975893760730643 T^{4} + \)\(13\!\cdots\!64\)\( T^{5} + \)\(40\!\cdots\!64\)\( T^{6} + \)\(13\!\cdots\!64\)\( p^{5} T^{7} + 4621975893760730643 p^{10} T^{8} + 97504017632694 p^{15} T^{9} + 3221213064 p^{20} T^{10} + 28874 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 - 972 T + 3101210150 T^{2} - 30836646344524 T^{3} + 4732064486300054663 T^{4} - \)\(58\!\cdots\!96\)\( T^{5} + \)\(48\!\cdots\!56\)\( T^{6} - \)\(58\!\cdots\!96\)\( p^{5} T^{7} + 4732064486300054663 p^{10} T^{8} - 30836646344524 p^{15} T^{9} + 3101210150 p^{20} T^{10} - 972 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 - 60040 T + 4014007166 T^{2} - 258076376751032 T^{3} + 12143787004248748327 T^{4} - \)\(51\!\cdots\!76\)\( T^{5} + \)\(21\!\cdots\!52\)\( T^{6} - \)\(51\!\cdots\!76\)\( p^{5} T^{7} + 12143787004248748327 p^{10} T^{8} - 258076376751032 p^{15} T^{9} + 4014007166 p^{20} T^{10} - 60040 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( 1 + 74174 T + 7442057520 T^{2} + 425960722537194 T^{3} + 27304246661399183355 T^{4} + \)\(12\!\cdots\!36\)\( T^{5} + \)\(85\!\cdots\!60\)\( p T^{6} + \)\(12\!\cdots\!36\)\( p^{5} T^{7} + 27304246661399183355 p^{10} T^{8} + 425960722537194 p^{15} T^{9} + 7442057520 p^{20} T^{10} + 74174 p^{25} T^{11} + p^{30} T^{12} \) | |
| 73 | \( 1 - 48960 T + 5754305510 T^{2} - 126338219241280 T^{3} + 8704246729227392447 T^{4} + \)\(12\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!20\)\( p^{5} T^{7} + 8704246729227392447 p^{10} T^{8} - 126338219241280 p^{15} T^{9} + 5754305510 p^{20} T^{10} - 48960 p^{25} T^{11} + p^{30} T^{12} \) | |
| 79 | \( 1 - 214508 T + 20515785810 T^{2} - 975267838794916 T^{3} + 3402012976766955919 T^{4} + \)\(30\!\cdots\!32\)\( T^{5} - \)\(30\!\cdots\!60\)\( p T^{6} + \)\(30\!\cdots\!32\)\( p^{5} T^{7} + 3402012976766955919 p^{10} T^{8} - 975267838794916 p^{15} T^{9} + 20515785810 p^{20} T^{10} - 214508 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 - 123260 T + 22129165854 T^{2} - 1831453215692004 T^{3} + \)\(19\!\cdots\!43\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{5} + \)\(10\!\cdots\!36\)\( T^{6} - \)\(12\!\cdots\!36\)\( p^{5} T^{7} + \)\(19\!\cdots\!43\)\( p^{10} T^{8} - 1831453215692004 p^{15} T^{9} + 22129165854 p^{20} T^{10} - 123260 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( 1 - 94028 T + 21807721730 T^{2} - 1775360494182844 T^{3} + \)\(23\!\cdots\!75\)\( T^{4} - \)\(16\!\cdots\!08\)\( T^{5} + \)\(16\!\cdots\!88\)\( T^{6} - \)\(16\!\cdots\!08\)\( p^{5} T^{7} + \)\(23\!\cdots\!75\)\( p^{10} T^{8} - 1775360494182844 p^{15} T^{9} + 21807721730 p^{20} T^{10} - 94028 p^{25} T^{11} + p^{30} T^{12} \) | |
| 97 | \( 1 - 246284 T + 51582760322 T^{2} - 6224284501236572 T^{3} + \)\(70\!\cdots\!39\)\( T^{4} - \)\(56\!\cdots\!20\)\( T^{5} + \)\(55\!\cdots\!56\)\( T^{6} - \)\(56\!\cdots\!20\)\( p^{5} T^{7} + \)\(70\!\cdots\!39\)\( p^{10} T^{8} - 6224284501236572 p^{15} T^{9} + 51582760322 p^{20} T^{10} - 246284 p^{25} T^{11} + p^{30} T^{12} \) | |
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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
−7.71964684036142276163564233258, −7.42400169915849450744015128847, −7.21930438897657010688068794210, −6.86566879628590454806447003936, −6.55139543087632280872520071166, −6.25316346749667806010363266921, −6.22712299221087313494738312615, −5.45404543885817578400423083346, −5.28768182067449187744975508917, −5.12159272704759478332638849424, −4.71899630250145764116388999436, −4.64829563190216059202619554769, −4.57601602086525222392157661926, −3.90113677682194073475758266731, −3.77361088440411179797917789331, −3.39033522329111310498604656107, −3.24015045185397993896592956832, −3.20507252859091977178613137695, −2.70551452962236041353648295606, −2.57252223625579979111211582909, −2.04451279006606430019326417420, −1.34248783356685451830123653619, −0.907854175455088691918292011102, −0.76699392355513239405568926471, −0.54027887011482223003609123981, 0.54027887011482223003609123981, 0.76699392355513239405568926471, 0.907854175455088691918292011102, 1.34248783356685451830123653619, 2.04451279006606430019326417420, 2.57252223625579979111211582909, 2.70551452962236041353648295606, 3.20507252859091977178613137695, 3.24015045185397993896592956832, 3.39033522329111310498604656107, 3.77361088440411179797917789331, 3.90113677682194073475758266731, 4.57601602086525222392157661926, 4.64829563190216059202619554769, 4.71899630250145764116388999436, 5.12159272704759478332638849424, 5.28768182067449187744975508917, 5.45404543885817578400423083346, 6.22712299221087313494738312615, 6.25316346749667806010363266921, 6.55139543087632280872520071166, 6.86566879628590454806447003936, 7.21930438897657010688068794210, 7.42400169915849450744015128847, 7.71964684036142276163564233258
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.