Dirichlet series
| L(s) = 1 | + 24·2-s − 36·3-s + 336·4-s − 94·5-s − 864·6-s − 194·7-s + 3.58e3·8-s + 177·9-s − 2.25e3·10-s − 258·11-s − 1.20e4·12-s − 2.37e3·13-s − 4.65e3·14-s + 3.38e3·15-s + 3.22e4·16-s + 862·17-s + 4.24e3·18-s − 3.22e3·19-s − 3.15e4·20-s + 6.98e3·21-s − 6.19e3·22-s − 1.29e4·23-s − 1.29e5·24-s − 7.21e3·25-s − 5.68e4·26-s + 8.24e3·27-s − 6.51e4·28-s + ⋯ |
| L(s) = 1 | + 4.24·2-s − 2.30·3-s + 21/2·4-s − 1.68·5-s − 9.79·6-s − 1.49·7-s + 19.7·8-s + 0.728·9-s − 7.13·10-s − 0.642·11-s − 24.2·12-s − 3.88·13-s − 6.34·14-s + 3.88·15-s + 63/2·16-s + 0.723·17-s + 3.09·18-s − 2.04·19-s − 17.6·20-s + 3.45·21-s − 2.72·22-s − 5.08·23-s − 45.7·24-s − 2.30·25-s − 16.5·26-s + 2.17·27-s − 15.7·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 79^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 79^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 79^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.64792\times 10^{8}\) |
| Root analytic conductor: | \(5.03394\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{6} \cdot 79^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{2} T )^{6} \) |
| 79 | \( ( 1 - p^{2} T )^{6} \) | |
| good | 3 | \( 1 + 4 p^{2} T + 373 p T^{2} + 25670 T^{3} + 17717 p^{3} T^{4} + 2712262 p T^{5} + 4793750 p^{3} T^{6} + 2712262 p^{6} T^{7} + 17717 p^{13} T^{8} + 25670 p^{15} T^{9} + 373 p^{21} T^{10} + 4 p^{27} T^{11} + p^{30} T^{12} \) |
| 5 | \( 1 + 94 T + 16052 T^{2} + 1115762 T^{3} + 110558973 T^{4} + 6062975794 T^{5} + 441084886899 T^{6} + 6062975794 p^{5} T^{7} + 110558973 p^{10} T^{8} + 1115762 p^{15} T^{9} + 16052 p^{20} T^{10} + 94 p^{25} T^{11} + p^{30} T^{12} \) | |
| 7 | \( 1 + 194 T + 51617 T^{2} + 9486156 T^{3} + 1649559979 T^{4} + 233452636202 T^{5} + 36013251872502 T^{6} + 233452636202 p^{5} T^{7} + 1649559979 p^{10} T^{8} + 9486156 p^{15} T^{9} + 51617 p^{20} T^{10} + 194 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 + 258 T + 482385 T^{2} + 48591714 T^{3} + 120934663954 T^{4} + 9126506927842 T^{5} + 23749353601084640 T^{6} + 9126506927842 p^{5} T^{7} + 120934663954 p^{10} T^{8} + 48591714 p^{15} T^{9} + 482385 p^{20} T^{10} + 258 p^{25} T^{11} + p^{30} T^{12} \) | |
| 13 | \( 1 + 2370 T + 20848 p^{2} T^{2} + 3572479922 T^{3} + 2911014198117 T^{4} + 152438730014926 p T^{5} + 1253192297695397255 T^{6} + 152438730014926 p^{6} T^{7} + 2911014198117 p^{10} T^{8} + 3572479922 p^{15} T^{9} + 20848 p^{22} T^{10} + 2370 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( 1 - 862 T + 4715786 T^{2} - 1673726518 T^{3} + 7345430797439 T^{4} + 1626595536612692 T^{5} + 7568814437142491564 T^{6} + 1626595536612692 p^{5} T^{7} + 7345430797439 p^{10} T^{8} - 1673726518 p^{15} T^{9} + 4715786 p^{20} T^{10} - 862 p^{25} T^{11} + p^{30} T^{12} \) | |
| 19 | \( 1 + 3222 T + 14764775 T^{2} + 30196192786 T^{3} + 80680803405518 T^{4} + 123469044328615908 T^{5} + \)\(24\!\cdots\!80\)\( T^{6} + 123469044328615908 p^{5} T^{7} + 80680803405518 p^{10} T^{8} + 30196192786 p^{15} T^{9} + 14764775 p^{20} T^{10} + 3222 p^{25} T^{11} + p^{30} T^{12} \) | |
| 23 | \( 1 + 12906 T + 99722715 T^{2} + 547648155782 T^{3} + 2330142561643950 T^{4} + 7964201417389776048 T^{5} + \)\(22\!\cdots\!28\)\( T^{6} + 7964201417389776048 p^{5} T^{7} + 2330142561643950 p^{10} T^{8} + 547648155782 p^{15} T^{9} + 99722715 p^{20} T^{10} + 12906 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 + 3400 T + 36437858 T^{2} + 169108137080 T^{3} + 1139298030309735 T^{4} + 4793215186882831840 T^{5} + \)\(31\!\cdots\!12\)\( T^{6} + 4793215186882831840 p^{5} T^{7} + 1139298030309735 p^{10} T^{8} + 169108137080 p^{15} T^{9} + 36437858 p^{20} T^{10} + 3400 p^{25} T^{11} + p^{30} T^{12} \) | |
| 31 | \( 1 + 6372 T + 59367441 T^{2} + 119236601506 T^{3} + 2058480353347138 T^{4} + 7937594243451997418 T^{5} + \)\(92\!\cdots\!04\)\( T^{6} + 7937594243451997418 p^{5} T^{7} + 2058480353347138 p^{10} T^{8} + 119236601506 p^{15} T^{9} + 59367441 p^{20} T^{10} + 6372 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( 1 - 4468 T + 70833446 T^{2} - 1901081175292 T^{3} + 9011427717118551 T^{4} - 91076677136410066144 T^{5} + \)\(15\!\cdots\!04\)\( T^{6} - 91076677136410066144 p^{5} T^{7} + 9011427717118551 p^{10} T^{8} - 1901081175292 p^{15} T^{9} + 70833446 p^{20} T^{10} - 4468 p^{25} T^{11} + p^{30} T^{12} \) | |
| 41 | \( 1 + 45328 T + 1427152686 T^{2} + 31236981165768 T^{3} + 553648734794728959 T^{4} + \)\(78\!\cdots\!56\)\( T^{5} + \)\(93\!\cdots\!12\)\( T^{6} + \)\(78\!\cdots\!56\)\( p^{5} T^{7} + 553648734794728959 p^{10} T^{8} + 31236981165768 p^{15} T^{9} + 1427152686 p^{20} T^{10} + 45328 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 + 10068 T + 30919142 T^{2} - 1184999739300 T^{3} + 3073336640579255 T^{4} - \)\(10\!\cdots\!32\)\( T^{5} - \)\(77\!\cdots\!96\)\( T^{6} - \)\(10\!\cdots\!32\)\( p^{5} T^{7} + 3073336640579255 p^{10} T^{8} - 1184999739300 p^{15} T^{9} + 30919142 p^{20} T^{10} + 10068 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 + 58924 T + 2606681909 T^{2} + 77188314444428 T^{3} + 1910407960056871387 T^{4} + \)\(37\!\cdots\!88\)\( T^{5} + \)\(62\!\cdots\!46\)\( T^{6} + \)\(37\!\cdots\!88\)\( p^{5} T^{7} + 1910407960056871387 p^{10} T^{8} + 77188314444428 p^{15} T^{9} + 2606681909 p^{20} T^{10} + 58924 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( 1 - 7126 T + 2171451294 T^{2} - 14361480576414 T^{3} + 2076633887281319399 T^{4} - \)\(11\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!44\)\( T^{6} - \)\(11\!\cdots\!68\)\( p^{5} T^{7} + 2076633887281319399 p^{10} T^{8} - 14361480576414 p^{15} T^{9} + 2171451294 p^{20} T^{10} - 7126 p^{25} T^{11} + p^{30} T^{12} \) | |
| 59 | \( 1 + 32414 T + 3473268063 T^{2} + 82330184459774 T^{3} + 5169985153462853967 T^{4} + \)\(94\!\cdots\!48\)\( T^{5} + \)\(45\!\cdots\!42\)\( T^{6} + \)\(94\!\cdots\!48\)\( p^{5} T^{7} + 5169985153462853967 p^{10} T^{8} + 82330184459774 p^{15} T^{9} + 3473268063 p^{20} T^{10} + 32414 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 - 36390 T + 1567537314 T^{2} - 58583983547598 T^{3} + 1673921654909480535 T^{4} - \)\(49\!\cdots\!68\)\( T^{5} + \)\(18\!\cdots\!28\)\( T^{6} - \)\(49\!\cdots\!68\)\( p^{5} T^{7} + 1673921654909480535 p^{10} T^{8} - 58583983547598 p^{15} T^{9} + 1567537314 p^{20} T^{10} - 36390 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 + 37880 T + 6722614411 T^{2} + 182417231460574 T^{3} + 19275105345573142594 T^{4} + \)\(39\!\cdots\!76\)\( T^{5} + \)\(32\!\cdots\!16\)\( T^{6} + \)\(39\!\cdots\!76\)\( p^{5} T^{7} + 19275105345573142594 p^{10} T^{8} + 182417231460574 p^{15} T^{9} + 6722614411 p^{20} T^{10} + 37880 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( 1 + 34396 T + 4733670585 T^{2} + 139051893543580 T^{3} + 14790649233318335095 T^{4} + \)\(39\!\cdots\!60\)\( T^{5} + \)\(32\!\cdots\!22\)\( T^{6} + \)\(39\!\cdots\!60\)\( p^{5} T^{7} + 14790649233318335095 p^{10} T^{8} + 139051893543580 p^{15} T^{9} + 4733670585 p^{20} T^{10} + 34396 p^{25} T^{11} + p^{30} T^{12} \) | |
| 73 | \( 1 - 70966 T + 9552928737 T^{2} - 432759737069932 T^{3} + 34145710811149431924 T^{4} - \)\(11\!\cdots\!98\)\( T^{5} + \)\(77\!\cdots\!48\)\( T^{6} - \)\(11\!\cdots\!98\)\( p^{5} T^{7} + 34145710811149431924 p^{10} T^{8} - 432759737069932 p^{15} T^{9} + 9552928737 p^{20} T^{10} - 70966 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 + 9330 T + 8237140610 T^{2} - 39667235158098 T^{3} + 40145999939592440647 T^{4} - \)\(85\!\cdots\!28\)\( T^{5} + \)\(13\!\cdots\!12\)\( T^{6} - \)\(85\!\cdots\!28\)\( p^{5} T^{7} + 40145999939592440647 p^{10} T^{8} - 39667235158098 p^{15} T^{9} + 8237140610 p^{20} T^{10} + 9330 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( 1 - 194380 T + 30444332774 T^{2} - 3748769186096806 T^{3} + \)\(39\!\cdots\!63\)\( T^{4} - \)\(35\!\cdots\!02\)\( T^{5} + \)\(27\!\cdots\!49\)\( T^{6} - \)\(35\!\cdots\!02\)\( p^{5} T^{7} + \)\(39\!\cdots\!63\)\( p^{10} T^{8} - 3748769186096806 p^{15} T^{9} + 30444332774 p^{20} T^{10} - 194380 p^{25} T^{11} + p^{30} T^{12} \) | |
| 97 | \( 1 + 93248 T + 32498406674 T^{2} + 2788225491453434 T^{3} + \)\(52\!\cdots\!03\)\( T^{4} + \)\(36\!\cdots\!62\)\( T^{5} + \)\(55\!\cdots\!73\)\( T^{6} + \)\(36\!\cdots\!62\)\( p^{5} T^{7} + \)\(52\!\cdots\!03\)\( p^{10} T^{8} + 2788225491453434 p^{15} T^{9} + 32498406674 p^{20} T^{10} + 93248 p^{25} T^{11} + p^{30} T^{12} \) | |
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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
−6.64369344787424608173536498718, −6.21540666457200248756720193046, −6.12269072356017555667023878577, −5.98463636577741140534024759561, −5.95090950074910639675943749082, −5.89396399560703645754137739386, −5.47926122028392963744159289780, −5.19068134188141296154220918391, −5.12543593230098323031431817359, −5.03369721507691135655869471394, −4.71173199163839998983899424061, −4.69947838052057728421248166103, −4.23041130366955365047680731198, −4.19083789985554962115715122200, −3.74432919863714383550993801535, −3.65440873970314037680443084777, −3.47450991086011621828021691267, −3.47130470679958738926274480913, −3.14590639227230608989492611348, −2.58521054149733918534828913536, −2.42283933479490481121082168039, −2.13790119162261416652076629261, −2.06780598717021587554843533770, −1.77930964062145825601790568000, −1.54900412336228360604458168361, 0, 0, 0, 0, 0, 0, 1.54900412336228360604458168361, 1.77930964062145825601790568000, 2.06780598717021587554843533770, 2.13790119162261416652076629261, 2.42283933479490481121082168039, 2.58521054149733918534828913536, 3.14590639227230608989492611348, 3.47130470679958738926274480913, 3.47450991086011621828021691267, 3.65440873970314037680443084777, 3.74432919863714383550993801535, 4.19083789985554962115715122200, 4.23041130366955365047680731198, 4.69947838052057728421248166103, 4.71173199163839998983899424061, 5.03369721507691135655869471394, 5.12543593230098323031431817359, 5.19068134188141296154220918391, 5.47926122028392963744159289780, 5.89396399560703645754137739386, 5.95090950074910639675943749082, 5.98463636577741140534024759561, 6.12269072356017555667023878577, 6.21540666457200248756720193046, 6.64369344787424608173536498718
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.