Dirichlet series
| L(s) = 1 | + 2·3-s − 75·5-s − 157·7-s − 164·9-s − 1.29e3·11-s + 244·13-s − 150·15-s − 1.19e3·17-s − 5.08e3·19-s − 314·21-s − 3.71e3·23-s − 6.27e3·25-s − 3.28e3·27-s − 9.49e3·29-s + 5.76e3·31-s − 2.59e3·33-s + 1.17e4·35-s − 3.34e4·37-s + 488·39-s − 2.05e4·41-s − 9.30e3·43-s + 1.23e4·45-s − 6.40e3·47-s − 2.48e4·49-s − 2.38e3·51-s − 3.12e4·53-s + 9.72e4·55-s + ⋯ |
| L(s) = 1 | + 0.128·3-s − 1.34·5-s − 1.21·7-s − 0.674·9-s − 3.22·11-s + 0.400·13-s − 0.172·15-s − 0.998·17-s − 3.22·19-s − 0.155·21-s − 1.46·23-s − 2.00·25-s − 0.866·27-s − 2.09·29-s + 1.07·31-s − 0.414·33-s + 1.62·35-s − 4.01·37-s + 0.0513·39-s − 1.91·41-s − 0.767·43-s + 0.905·45-s − 0.423·47-s − 1.47·49-s − 0.128·51-s − 1.52·53-s + 4.33·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{12} \cdot 31^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.18718\times 10^{7}\) |
| Root analytic conductor: | \(4.45955\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{12} \cdot 31^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
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 | \( 1 \) |
| 31 | \( ( 1 - p^{2} T )^{6} \) | |
| good | 3 | \( 1 - 2 T + 56 p T^{2} + 2618 T^{3} - 42577 T^{4} - 198212 p T^{5} - 713536 p^{2} T^{6} - 198212 p^{6} T^{7} - 42577 p^{10} T^{8} + 2618 p^{15} T^{9} + 56 p^{21} T^{10} - 2 p^{25} T^{11} + p^{30} T^{12} \) |
| 5 | \( 1 + 3 p^{2} T + 11903 T^{2} + 768288 T^{3} + 70952053 T^{4} + 3995966037 T^{5} + 271192478518 T^{6} + 3995966037 p^{5} T^{7} + 70952053 p^{10} T^{8} + 768288 p^{15} T^{9} + 11903 p^{20} T^{10} + 3 p^{27} T^{11} + p^{30} T^{12} \) | |
| 7 | \( 1 + 157 T + 7067 p T^{2} + 3393794 T^{3} + 705222435 T^{4} - 18157806027 T^{5} + 103159459038 p^{2} T^{6} - 18157806027 p^{5} T^{7} + 705222435 p^{10} T^{8} + 3393794 p^{15} T^{9} + 7067 p^{21} T^{10} + 157 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 + 1296 T + 1126560 T^{2} + 773765100 T^{3} + 449591439255 T^{4} + 221533644893868 T^{5} + 95365079688566752 T^{6} + 221533644893868 p^{5} T^{7} + 449591439255 p^{10} T^{8} + 773765100 p^{15} T^{9} + 1126560 p^{20} T^{10} + 1296 p^{25} T^{11} + p^{30} T^{12} \) | |
| 13 | \( 1 - 244 T + 1056496 T^{2} - 31373776 p T^{3} + 587845822431 T^{4} - 240676770285084 T^{5} + 249603968909628816 T^{6} - 240676770285084 p^{5} T^{7} + 587845822431 p^{10} T^{8} - 31373776 p^{16} T^{9} + 1056496 p^{20} T^{10} - 244 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( 1 + 70 p T + 267690 p T^{2} + 93100366 p T^{3} + 6514677950223 T^{4} - 3188027843473012 T^{5} + 6400797664772583052 T^{6} - 3188027843473012 p^{5} T^{7} + 6514677950223 p^{10} T^{8} + 93100366 p^{16} T^{9} + 267690 p^{21} T^{10} + 70 p^{26} T^{11} + p^{30} T^{12} \) | |
| 19 | \( 1 + 5081 T + 21365105 T^{2} + 59447171322 T^{3} + 146587910691991 T^{4} + 284349946917009397 T^{5} + \)\(49\!\cdots\!90\)\( T^{6} + 284349946917009397 p^{5} T^{7} + 146587910691991 p^{10} T^{8} + 59447171322 p^{15} T^{9} + 21365105 p^{20} T^{10} + 5081 p^{25} T^{11} + p^{30} T^{12} \) | |
| 23 | \( 1 + 3716 T + 28240002 T^{2} + 80651119220 T^{3} + 391430465303967 T^{4} + 889992662182973168 T^{5} + \)\(31\!\cdots\!40\)\( T^{6} + 889992662182973168 p^{5} T^{7} + 391430465303967 p^{10} T^{8} + 80651119220 p^{15} T^{9} + 28240002 p^{20} T^{10} + 3716 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 + 9494 T + 130138500 T^{2} + 847876724894 T^{3} + 6600490485687231 T^{4} + 32021815000523224772 T^{5} + \)\(61\!\cdots\!24\)\( p T^{6} + 32021815000523224772 p^{5} T^{7} + 6600490485687231 p^{10} T^{8} + 847876724894 p^{15} T^{9} + 130138500 p^{20} T^{10} + 9494 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( 1 + 33418 T + 773910836 T^{2} + 12412139736002 T^{3} + 163285454004924255 T^{4} + \)\(17\!\cdots\!52\)\( T^{5} + \)\(15\!\cdots\!28\)\( T^{6} + \)\(17\!\cdots\!52\)\( p^{5} T^{7} + 163285454004924255 p^{10} T^{8} + 12412139736002 p^{15} T^{9} + 773910836 p^{20} T^{10} + 33418 p^{25} T^{11} + p^{30} T^{12} \) | |
| 41 | \( 1 + 20581 T + 713766711 T^{2} + 10466516696056 T^{3} + 204860673581481645 T^{4} + \)\(22\!\cdots\!87\)\( T^{5} + \)\(31\!\cdots\!18\)\( T^{6} + \)\(22\!\cdots\!87\)\( p^{5} T^{7} + 204860673581481645 p^{10} T^{8} + 10466516696056 p^{15} T^{9} + 713766711 p^{20} T^{10} + 20581 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 + 9304 T + 618558024 T^{2} + 5493313414692 T^{3} + 186762491827026199 T^{4} + \)\(14\!\cdots\!04\)\( T^{5} + \)\(34\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!04\)\( p^{5} T^{7} + 186762491827026199 p^{10} T^{8} + 5493313414692 p^{15} T^{9} + 618558024 p^{20} T^{10} + 9304 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 + 6408 T + 937247678 T^{2} + 4444537326696 T^{3} + 430602179765219359 T^{4} + \)\(15\!\cdots\!84\)\( T^{5} + \)\(12\!\cdots\!28\)\( T^{6} + \)\(15\!\cdots\!84\)\( p^{5} T^{7} + 430602179765219359 p^{10} T^{8} + 4444537326696 p^{15} T^{9} + 937247678 p^{20} T^{10} + 6408 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( 1 + 31284 T + 206451632 T^{2} - 7552601977344 T^{3} - 23301926994107825 T^{4} + \)\(39\!\cdots\!76\)\( T^{5} + \)\(13\!\cdots\!88\)\( T^{6} + \)\(39\!\cdots\!76\)\( p^{5} T^{7} - 23301926994107825 p^{10} T^{8} - 7552601977344 p^{15} T^{9} + 206451632 p^{20} T^{10} + 31284 p^{25} T^{11} + p^{30} T^{12} \) | |
| 59 | \( 1 + 41039 T + 2851209309 T^{2} + 94701529644470 T^{3} + 4127554549079245419 T^{4} + \)\(11\!\cdots\!67\)\( T^{5} + \)\(36\!\cdots\!70\)\( T^{6} + \)\(11\!\cdots\!67\)\( p^{5} T^{7} + 4127554549079245419 p^{10} T^{8} + 94701529644470 p^{15} T^{9} + 2851209309 p^{20} T^{10} + 41039 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 - 58788 T + 4372440848 T^{2} - 193343963563376 T^{3} + 7975771686202553663 T^{4} - \)\(28\!\cdots\!12\)\( T^{5} + \)\(84\!\cdots\!52\)\( T^{6} - \)\(28\!\cdots\!12\)\( p^{5} T^{7} + 7975771686202553663 p^{10} T^{8} - 193343963563376 p^{15} T^{9} + 4372440848 p^{20} T^{10} - 58788 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 + 65432 T + 4537924162 T^{2} + 147249789434984 T^{3} + 4594583077221647655 T^{4} + \)\(51\!\cdots\!32\)\( T^{5} + \)\(15\!\cdots\!60\)\( T^{6} + \)\(51\!\cdots\!32\)\( p^{5} T^{7} + 4594583077221647655 p^{10} T^{8} + 147249789434984 p^{15} T^{9} + 4537924162 p^{20} T^{10} + 65432 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( 1 - 15361 T + 4026394173 T^{2} - 153223150403422 T^{3} + 10835483462684528811 T^{4} - \)\(53\!\cdots\!13\)\( T^{5} + \)\(19\!\cdots\!18\)\( T^{6} - \)\(53\!\cdots\!13\)\( p^{5} T^{7} + 10835483462684528811 p^{10} T^{8} - 153223150403422 p^{15} T^{9} + 4026394173 p^{20} T^{10} - 15361 p^{25} T^{11} + p^{30} T^{12} \) | |
| 73 | \( 1 + 35468 T + 4940503186 T^{2} + 80191430879228 T^{3} + 6839332776316546143 T^{4} - \)\(24\!\cdots\!00\)\( T^{5} + \)\(35\!\cdots\!12\)\( T^{6} - \)\(24\!\cdots\!00\)\( p^{5} T^{7} + 6839332776316546143 p^{10} T^{8} + 80191430879228 p^{15} T^{9} + 4940503186 p^{20} T^{10} + 35468 p^{25} T^{11} + p^{30} T^{12} \) | |
| 79 | \( 1 + 12722 T + 13202937742 T^{2} + 78848517848246 T^{3} + 78765155220530061951 T^{4} + \)\(16\!\cdots\!28\)\( T^{5} + \)\(29\!\cdots\!12\)\( T^{6} + \)\(16\!\cdots\!28\)\( p^{5} T^{7} + 78765155220530061951 p^{10} T^{8} + 78848517848246 p^{15} T^{9} + 13202937742 p^{20} T^{10} + 12722 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 + 76064 T + 16589654584 T^{2} + 877650836833844 T^{3} + \)\(11\!\cdots\!07\)\( T^{4} + \)\(45\!\cdots\!44\)\( T^{5} + \)\(50\!\cdots\!36\)\( T^{6} + \)\(45\!\cdots\!44\)\( p^{5} T^{7} + \)\(11\!\cdots\!07\)\( p^{10} T^{8} + 877650836833844 p^{15} T^{9} + 16589654584 p^{20} T^{10} + 76064 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( 1 - 140176 T + 34676680554 T^{2} - 3629885102415544 T^{3} + \)\(49\!\cdots\!47\)\( T^{4} - \)\(39\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!32\)\( T^{6} - \)\(39\!\cdots\!00\)\( p^{5} T^{7} + \)\(49\!\cdots\!47\)\( p^{10} T^{8} - 3629885102415544 p^{15} T^{9} + 34676680554 p^{20} T^{10} - 140176 p^{25} T^{11} + p^{30} T^{12} \) | |
| 97 | \( 1 - 197177 T + 52132907435 T^{2} - 7406790039920032 T^{3} + \)\(11\!\cdots\!77\)\( T^{4} - \)\(11\!\cdots\!67\)\( T^{5} + \)\(12\!\cdots\!70\)\( T^{6} - \)\(11\!\cdots\!67\)\( p^{5} T^{7} + \)\(11\!\cdots\!77\)\( p^{10} T^{8} - 7406790039920032 p^{15} T^{9} + 52132907435 p^{20} T^{10} - 197177 p^{25} T^{11} + p^{30} T^{12} \) | |
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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
−7.02368880342641902133293154559, −7.00785168770995271861111620939, −6.58448958410387631465509898712, −6.24924205496904580077646873964, −6.24005531099535466954892729585, −6.22783778809995794472830407089, −6.14629012081359968033559089822, −5.51057445713540854757071562857, −5.33109260166417331523356447561, −5.16950921299098673748245888664, −4.99956798220850967816774964501, −4.85772541347101864095203376368, −4.42823104467903769819064747932, −4.14800634989971588217056959218, −3.96928503981346133895583755383, −3.55395376944929084729243396403, −3.55053579800261356112642434512, −3.44213754202371515476130139585, −3.15578404778014092921353424625, −2.53451922308516065276482963509, −2.38915119929079954363371576171, −2.17660581711928557098532433330, −2.05378378433527642103298201338, −1.53532569528066776070178442075, −1.47908546412689382354548337108, 0, 0, 0, 0, 0, 0, 1.47908546412689382354548337108, 1.53532569528066776070178442075, 2.05378378433527642103298201338, 2.17660581711928557098532433330, 2.38915119929079954363371576171, 2.53451922308516065276482963509, 3.15578404778014092921353424625, 3.44213754202371515476130139585, 3.55053579800261356112642434512, 3.55395376944929084729243396403, 3.96928503981346133895583755383, 4.14800634989971588217056959218, 4.42823104467903769819064747932, 4.85772541347101864095203376368, 4.99956798220850967816774964501, 5.16950921299098673748245888664, 5.33109260166417331523356447561, 5.51057445713540854757071562857, 6.14629012081359968033559089822, 6.22783778809995794472830407089, 6.24005531099535466954892729585, 6.24924205496904580077646873964, 6.58448958410387631465509898712, 7.00785168770995271861111620939, 7.02368880342641902133293154559
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.