Dirichlet series
| L(s) = 1 | + 24·3-s + 12·4-s + 360·9-s + 288·12-s + 48·16-s − 180·17-s + 38·23-s + 872·25-s + 4.37e3·27-s + 202·29-s + 4.32e3·36-s − 2.41e3·43-s + 1.15e3·48-s − 1.21e3·49-s − 4.32e3·51-s − 4.38e3·53-s + 2.21e3·61-s − 128·64-s − 2.16e3·68-s + 912·69-s + 2.09e4·75-s − 7.84e3·79-s + 4.49e4·81-s + 4.84e3·87-s + 456·92-s + 1.04e4·100-s + 2.55e3·101-s + ⋯ |
| L(s) = 1 | + 4.61·3-s + 3/2·4-s + 40/3·9-s + 6.92·12-s + 3/4·16-s − 2.56·17-s + 0.344·23-s + 6.97·25-s + 31.1·27-s + 1.29·29-s + 20·36-s − 8.54·43-s + 3.46·48-s − 3.53·49-s − 11.8·51-s − 11.3·53-s + 4.65·61-s − 1/4·64-s − 3.85·68-s + 1.59·69-s + 32.2·75-s − 11.1·79-s + 61.6·81-s + 5.97·87-s + 0.516·92-s + 10.4·100-s + 2.51·101-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 13^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.95724\times 10^{15}\) |
| Root analytic conductor: | \(4.46571\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 13^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(8.519938782\) |
| \(L(\frac12)\) | \(\approx\) | \(8.519938782\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{3} \) |
| 13 | \( 1 \) | |
| good | 3 | \( ( 1 - 4 p T + 4 p^{2} T^{2} - 26 T^{3} + 376 p T^{4} - 272 p^{3} T^{5} + 22687 T^{6} - 272 p^{6} T^{7} + 376 p^{7} T^{8} - 26 p^{9} T^{9} + 4 p^{14} T^{10} - 4 p^{16} T^{11} + p^{18} T^{12} )^{2} \) |
| 5 | \( ( 1 - 436 T^{2} + 103056 T^{4} - 15453889 T^{6} + 103056 p^{6} T^{8} - 436 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 7 | \( 1 + 1213 T^{2} + 846032 T^{4} + 44311205 p T^{6} + 45548573507 T^{8} - 21505406726124 T^{10} - 12859398267661623 T^{12} - 21505406726124 p^{6} T^{14} + 45548573507 p^{12} T^{16} + 44311205 p^{19} T^{18} + 846032 p^{24} T^{20} + 1213 p^{30} T^{22} + p^{36} T^{24} \) | |
| 11 | \( 1 + 2940 T^{2} + 5572564 T^{4} + 4606009814 T^{6} - 2510834452120 T^{8} - 15971978573781272 T^{10} - 27825171402513079525 T^{12} - 15971978573781272 p^{6} T^{14} - 2510834452120 p^{12} T^{16} + 4606009814 p^{18} T^{18} + 5572564 p^{24} T^{20} + 2940 p^{30} T^{22} + p^{36} T^{24} \) | |
| 17 | \( ( 1 + 90 T - 5468 T^{2} - 393226 T^{3} + 44785470 T^{4} + 1125379354 T^{5} - 211247539689 T^{6} + 1125379354 p^{3} T^{7} + 44785470 p^{6} T^{8} - 393226 p^{9} T^{9} - 5468 p^{12} T^{10} + 90 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 19 | \( 1 + 15916 T^{2} + 202761708 T^{4} + 1133695405670 T^{6} + 2795908662819632 T^{8} - 48803075506997114688 T^{10} - \)\(44\!\cdots\!57\)\( T^{12} - 48803075506997114688 p^{6} T^{14} + 2795908662819632 p^{12} T^{16} + 1133695405670 p^{18} T^{18} + 202761708 p^{24} T^{20} + 15916 p^{30} T^{22} + p^{36} T^{24} \) | |
| 23 | \( ( 1 - 19 T - 17379 T^{2} + 1795946 T^{3} + 78601463 T^{4} - 13530427071 T^{5} + 857245646806 T^{6} - 13530427071 p^{3} T^{7} + 78601463 p^{6} T^{8} + 1795946 p^{9} T^{9} - 17379 p^{12} T^{10} - 19 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 29 | \( ( 1 - 101 T + 9670 T^{2} + 2392827 T^{3} - 136214727 T^{4} - 58110677102 T^{5} + 24435911693829 T^{6} - 58110677102 p^{3} T^{7} - 136214727 p^{6} T^{8} + 2392827 p^{9} T^{9} + 9670 p^{12} T^{10} - 101 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 31 | \( ( 1 - 67693 T^{2} + 2813159873 T^{4} - 88764051088713 T^{6} + 2813159873 p^{6} T^{8} - 67693 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 37 | \( 1 + 289120 T^{2} + 48092940392 T^{4} + 5518178251433198 T^{6} + \)\(48\!\cdots\!72\)\( T^{8} + \)\(33\!\cdots\!88\)\( T^{10} + \)\(18\!\cdots\!43\)\( T^{12} + \)\(33\!\cdots\!88\)\( p^{6} T^{14} + \)\(48\!\cdots\!72\)\( p^{12} T^{16} + 5518178251433198 p^{18} T^{18} + 48092940392 p^{24} T^{20} + 289120 p^{30} T^{22} + p^{36} T^{24} \) | |
| 41 | \( 1 + 72459 T^{2} - 2942996285 T^{4} - 784013353069900 T^{6} - 21701918099899481623 T^{8} + \)\(16\!\cdots\!73\)\( T^{10} + \)\(25\!\cdots\!94\)\( T^{12} + \)\(16\!\cdots\!73\)\( p^{6} T^{14} - 21701918099899481623 p^{12} T^{16} - 784013353069900 p^{18} T^{18} - 2942996285 p^{24} T^{20} + 72459 p^{30} T^{22} + p^{36} T^{24} \) | |
| 43 | \( ( 1 + 1205 T + 731136 T^{2} + 357194599 T^{3} + 154200549383 T^{4} + 53845297050762 T^{5} + 15910224576330043 T^{6} + 53845297050762 p^{3} T^{7} + 154200549383 p^{6} T^{8} + 357194599 p^{9} T^{9} + 731136 p^{12} T^{10} + 1205 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 47 | \( ( 1 - 184924 T^{2} + 13473005240 T^{4} - 366518815821141 T^{6} + 13473005240 p^{6} T^{8} - 184924 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 53 | \( ( 1 + 1095 T + 839313 T^{2} + 371911379 T^{3} + 839313 p^{3} T^{4} + 1095 p^{6} T^{5} + p^{9} T^{6} )^{4} \) | |
| 59 | \( 1 + 412807 T^{2} + 70300999051 T^{4} + 4687605950834692 T^{6} - \)\(17\!\cdots\!43\)\( T^{8} - \)\(91\!\cdots\!79\)\( T^{10} - \)\(24\!\cdots\!82\)\( T^{12} - \)\(91\!\cdots\!79\)\( p^{6} T^{14} - \)\(17\!\cdots\!43\)\( p^{12} T^{16} + 4687605950834692 p^{18} T^{18} + 70300999051 p^{24} T^{20} + 412807 p^{30} T^{22} + p^{36} T^{24} \) | |
| 61 | \( ( 1 - 1108 T + 195280 T^{2} - 81146578 T^{3} + 242730039476 T^{4} - 86381104403572 T^{5} + 569801340677051 T^{6} - 86381104403572 p^{3} T^{7} + 242730039476 p^{6} T^{8} - 81146578 p^{9} T^{9} + 195280 p^{12} T^{10} - 1108 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( 1 + 754705 T^{2} + 248140267820 T^{4} + 44550675395447459 T^{6} - \)\(48\!\cdots\!85\)\( T^{8} - \)\(58\!\cdots\!80\)\( T^{10} - \)\(27\!\cdots\!43\)\( T^{12} - \)\(58\!\cdots\!80\)\( p^{6} T^{14} - \)\(48\!\cdots\!85\)\( p^{12} T^{16} + 44550675395447459 p^{18} T^{18} + 248140267820 p^{24} T^{20} + 754705 p^{30} T^{22} + p^{36} T^{24} \) | |
| 71 | \( 1 + 1203945 T^{2} + 861630964656 T^{4} + 299768879179299207 T^{6} + \)\(24\!\cdots\!07\)\( T^{8} - \)\(41\!\cdots\!84\)\( T^{10} - \)\(22\!\cdots\!83\)\( T^{12} - \)\(41\!\cdots\!84\)\( p^{6} T^{14} + \)\(24\!\cdots\!07\)\( p^{12} T^{16} + 299768879179299207 p^{18} T^{18} + 861630964656 p^{24} T^{20} + 1203945 p^{30} T^{22} + p^{36} T^{24} \) | |
| 73 | \( ( 1 - 578397 T^{2} + 340509020657 T^{4} - 175733770047454529 T^{6} + 340509020657 p^{6} T^{8} - 578397 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 79 | \( ( 1 + 1961 T + 2641773 T^{2} + 2147763857 T^{3} + 2641773 p^{3} T^{4} + 1961 p^{6} T^{5} + p^{9} T^{6} )^{4} \) | |
| 83 | \( ( 1 - 3126181 T^{2} + 4236994280213 T^{4} - 3174148229464390881 T^{6} + 4236994280213 p^{6} T^{8} - 3126181 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 89 | \( 1 + 3632513 T^{2} + 7360263332404 T^{4} + 10561614224132377739 T^{6} + \)\(11\!\cdots\!71\)\( T^{8} + \)\(10\!\cdots\!28\)\( T^{10} + \)\(83\!\cdots\!97\)\( T^{12} + \)\(10\!\cdots\!28\)\( p^{6} T^{14} + \)\(11\!\cdots\!71\)\( p^{12} T^{16} + 10561614224132377739 p^{18} T^{18} + 7360263332404 p^{24} T^{20} + 3632513 p^{30} T^{22} + p^{36} T^{24} \) | |
| 97 | \( 1 + 3266797 T^{2} + 5378593419936 T^{4} + 5916317576309886611 T^{6} + \)\(50\!\cdots\!67\)\( T^{8} + \)\(35\!\cdots\!84\)\( T^{10} + \)\(27\!\cdots\!93\)\( T^{12} + \)\(35\!\cdots\!84\)\( p^{6} T^{14} + \)\(50\!\cdots\!67\)\( p^{12} T^{16} + 5916317576309886611 p^{18} T^{18} + 5378593419936 p^{24} T^{20} + 3266797 p^{30} T^{22} + p^{36} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.26497898307573854396602347696, −3.25961829442592949411396701286, −3.03000248180092872780894719604, −2.98532768305944774114661758752, −2.98105278733210415336166583976, −2.96739015943463796250869315628, −2.71646049329439524874690439539, −2.59856449349083383252209087232, −2.57754250324296734544808974412, −2.51670539696914686524641413787, −2.30101256485813820157234829878, −2.12550085636595206337915687446, −1.85387196551216425189324708488, −1.69789291561701999109720965013, −1.68115457801384361916567776499, −1.57672033931221252196939285676, −1.52852970341677321894936732830, −1.33230835016512939445078442583, −1.32575760091362613998215881256, −1.30188328656520887090271364335, −1.23486323533708843593078024304, −0.72501828456551874048616655719, −0.52383535353422078080396923563, −0.15867713882680696411996591993, −0.07303609752524847649026298262, 0.07303609752524847649026298262, 0.15867713882680696411996591993, 0.52383535353422078080396923563, 0.72501828456551874048616655719, 1.23486323533708843593078024304, 1.30188328656520887090271364335, 1.32575760091362613998215881256, 1.33230835016512939445078442583, 1.52852970341677321894936732830, 1.57672033931221252196939285676, 1.68115457801384361916567776499, 1.69789291561701999109720965013, 1.85387196551216425189324708488, 2.12550085636595206337915687446, 2.30101256485813820157234829878, 2.51670539696914686524641413787, 2.57754250324296734544808974412, 2.59856449349083383252209087232, 2.71646049329439524874690439539, 2.96739015943463796250869315628, 2.98105278733210415336166583976, 2.98532768305944774114661758752, 3.03000248180092872780894719604, 3.25961829442592949411396701286, 3.26497898307573854396602347696
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.