Dirichlet series
| L(s) = 1 | − 8·2-s − 8·4-s + 224·8-s − 180·9-s − 8·13-s − 302·16-s + 1.44e3·18-s − 352·19-s − 996·25-s + 64·26-s − 2.39e3·32-s + 1.44e3·36-s + 2.81e3·38-s − 1.20e3·43-s − 1.51e3·47-s − 2.58e3·49-s + 7.96e3·50-s + 64·52-s − 2.50e3·53-s − 3.40e3·59-s + 5.67e3·64-s − 1.08e3·67-s − 4.03e4·72-s + 2.81e3·76-s + 1.58e4·81-s − 2.96e3·83-s + 9.60e3·86-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 4-s + 9.89·8-s − 6.66·9-s − 0.170·13-s − 4.71·16-s + 18.8·18-s − 4.25·19-s − 7.96·25-s + 0.482·26-s − 13.2·32-s + 20/3·36-s + 12.0·38-s − 4.25·43-s − 4.69·47-s − 7.53·49-s + 22.5·50-s + 0.170·52-s − 6.48·53-s − 7.52·59-s + 11.0·64-s − 1.96·67-s − 65.9·72-s + 4.25·76-s + 21.7·81-s − 3.91·83-s + 12.0·86-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(17^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.04180\times 10^{14}\) |
| Root analytic conductor: | \(4.12935\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 17^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( ( 1 + p^{2} T + 7 p^{2} T^{2} + 3 p^{5} T^{3} + 431 T^{4} + 39 p^{5} T^{5} + 535 p^{3} T^{6} + 39 p^{8} T^{7} + 431 p^{6} T^{8} + 3 p^{14} T^{9} + 7 p^{14} T^{10} + p^{17} T^{11} + p^{18} T^{12} )^{2} \) |
| 3 | \( 1 + 20 p^{2} T^{2} + 16558 T^{4} + 338044 p T^{6} + 15420401 p T^{8} + 1671212744 T^{10} + 49530968732 T^{12} + 1671212744 p^{6} T^{14} + 15420401 p^{13} T^{16} + 338044 p^{19} T^{18} + 16558 p^{24} T^{20} + 20 p^{32} T^{22} + p^{36} T^{24} \) | |
| 5 | \( 1 + 996 T^{2} + 464566 T^{4} + 135566596 T^{6} + 28189444083 T^{8} + 4567447110728 T^{10} + 616822423856428 T^{12} + 4567447110728 p^{6} T^{14} + 28189444083 p^{12} T^{16} + 135566596 p^{18} T^{18} + 464566 p^{24} T^{20} + 996 p^{30} T^{22} + p^{36} T^{24} \) | |
| 7 | \( 1 + 2584 T^{2} + 477436 p T^{4} + 2854552536 T^{6} + 1790890578487 T^{8} + 868515387409360 T^{10} + 333557528272102152 T^{12} + 868515387409360 p^{6} T^{14} + 1790890578487 p^{12} T^{16} + 2854552536 p^{18} T^{18} + 477436 p^{25} T^{20} + 2584 p^{30} T^{22} + p^{36} T^{24} \) | |
| 11 | \( 1 + 7092 T^{2} + 25644206 T^{4} + 64055676788 T^{6} + 127150115092339 T^{8} + 213769156637131400 T^{10} + \)\(30\!\cdots\!40\)\( T^{12} + 213769156637131400 p^{6} T^{14} + 127150115092339 p^{12} T^{16} + 64055676788 p^{18} T^{18} + 25644206 p^{24} T^{20} + 7092 p^{30} T^{22} + p^{36} T^{24} \) | |
| 13 | \( ( 1 + 4 T + 7232 T^{2} + 32764 T^{3} + 27813463 T^{4} + 3908432 p T^{5} + 73018628704 T^{6} + 3908432 p^{4} T^{7} + 27813463 p^{6} T^{8} + 32764 p^{9} T^{9} + 7232 p^{12} T^{10} + 4 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 19 | \( ( 1 + 176 T + 34040 T^{2} + 4244352 T^{3} + 479644127 T^{4} + 47833890256 T^{5} + 4059366462720 T^{6} + 47833890256 p^{3} T^{7} + 479644127 p^{6} T^{8} + 4244352 p^{9} T^{9} + 34040 p^{12} T^{10} + 176 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 23 | \( 1 + 68952 T^{2} + 2200763172 T^{4} + 41413173138040 T^{6} + 480390044058280311 T^{8} + \)\(33\!\cdots\!00\)\( T^{10} + \)\(23\!\cdots\!56\)\( T^{12} + \)\(33\!\cdots\!00\)\( p^{6} T^{14} + 480390044058280311 p^{12} T^{16} + 41413173138040 p^{18} T^{18} + 2200763172 p^{24} T^{20} + 68952 p^{30} T^{22} + p^{36} T^{24} \) | |
| 29 | \( 1 + 133956 T^{2} + 9048121366 T^{4} + 425191427795748 T^{6} + 15699249682181213011 T^{8} + \)\(47\!\cdots\!12\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{12} + \)\(47\!\cdots\!12\)\( p^{6} T^{14} + 15699249682181213011 p^{12} T^{16} + 425191427795748 p^{18} T^{18} + 9048121366 p^{24} T^{20} + 133956 p^{30} T^{22} + p^{36} T^{24} \) | |
| 31 | \( 1 + 259304 T^{2} + 33052278564 T^{4} + 2715804861054696 T^{6} + \)\(15\!\cdots\!79\)\( T^{8} + \)\(70\!\cdots\!76\)\( T^{10} + \)\(23\!\cdots\!36\)\( T^{12} + \)\(70\!\cdots\!76\)\( p^{6} T^{14} + \)\(15\!\cdots\!79\)\( p^{12} T^{16} + 2715804861054696 p^{18} T^{18} + 33052278564 p^{24} T^{20} + 259304 p^{30} T^{22} + p^{36} T^{24} \) | |
| 37 | \( 1 + 326260 T^{2} + 55061041270 T^{4} + 6229097877608916 T^{6} + \)\(52\!\cdots\!31\)\( T^{8} + \)\(35\!\cdots\!04\)\( T^{10} + \)\(19\!\cdots\!60\)\( T^{12} + \)\(35\!\cdots\!04\)\( p^{6} T^{14} + \)\(52\!\cdots\!31\)\( p^{12} T^{16} + 6229097877608916 p^{18} T^{18} + 55061041270 p^{24} T^{20} + 326260 p^{30} T^{22} + p^{36} T^{24} \) | |
| 41 | \( 1 + 534912 T^{2} + 134299786720 T^{4} + 21276082554472832 T^{6} + \)\(24\!\cdots\!59\)\( T^{8} + \)\(21\!\cdots\!44\)\( T^{10} + \)\(16\!\cdots\!28\)\( T^{12} + \)\(21\!\cdots\!44\)\( p^{6} T^{14} + \)\(24\!\cdots\!59\)\( p^{12} T^{16} + 21276082554472832 p^{18} T^{18} + 134299786720 p^{24} T^{20} + 534912 p^{30} T^{22} + p^{36} T^{24} \) | |
| 43 | \( ( 1 + 600 T + 440350 T^{2} + 199103888 T^{3} + 85161550035 T^{4} + 29019164315880 T^{5} + 8954285815797980 T^{6} + 29019164315880 p^{3} T^{7} + 85161550035 p^{6} T^{8} + 199103888 p^{9} T^{9} + 440350 p^{12} T^{10} + 600 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 47 | \( ( 1 + 756 T + 575974 T^{2} + 273649852 T^{3} + 132395477023 T^{4} + 48206958866376 T^{5} + 17738585915593652 T^{6} + 48206958866376 p^{3} T^{7} + 132395477023 p^{6} T^{8} + 273649852 p^{9} T^{9} + 575974 p^{12} T^{10} + 756 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 53 | \( ( 1 + 1252 T + 1176134 T^{2} + 802951332 T^{3} + 460747112091 T^{4} + 220552256296280 T^{5} + 91961321435374060 T^{6} + 220552256296280 p^{3} T^{7} + 460747112091 p^{6} T^{8} + 802951332 p^{9} T^{9} + 1176134 p^{12} T^{10} + 1252 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 59 | \( ( 1 + 1704 T + 2004310 T^{2} + 1683295744 T^{3} + 1153507905267 T^{4} + 656384255916360 T^{5} + 321205504810995020 T^{6} + 656384255916360 p^{3} T^{7} + 1153507905267 p^{6} T^{8} + 1683295744 p^{9} T^{9} + 2004310 p^{12} T^{10} + 1704 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 61 | \( 1 + 1068596 T^{2} + 594842810678 T^{4} + 242972838259538900 T^{6} + \)\(80\!\cdots\!11\)\( T^{8} + \)\(22\!\cdots\!80\)\( T^{10} + \)\(55\!\cdots\!04\)\( T^{12} + \)\(22\!\cdots\!80\)\( p^{6} T^{14} + \)\(80\!\cdots\!11\)\( p^{12} T^{16} + 242972838259538900 p^{18} T^{18} + 594842810678 p^{24} T^{20} + 1068596 p^{30} T^{22} + p^{36} T^{24} \) | |
| 67 | \( ( 1 + 540 T + 1237196 T^{2} + 437516948 T^{3} + 687431155711 T^{4} + 191267075797288 T^{5} + 250196846973981368 T^{6} + 191267075797288 p^{3} T^{7} + 687431155711 p^{6} T^{8} + 437516948 p^{9} T^{9} + 1237196 p^{12} T^{10} + 540 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 71 | \( 1 + 3456584 T^{2} + 5689050756996 T^{4} + 5900580964453833256 T^{6} + \)\(42\!\cdots\!11\)\( T^{8} + \)\(23\!\cdots\!76\)\( T^{10} + \)\(94\!\cdots\!04\)\( T^{12} + \)\(23\!\cdots\!76\)\( p^{6} T^{14} + \)\(42\!\cdots\!11\)\( p^{12} T^{16} + 5900580964453833256 p^{18} T^{18} + 5689050756996 p^{24} T^{20} + 3456584 p^{30} T^{22} + p^{36} T^{24} \) | |
| 73 | \( 1 + 3427968 T^{2} + 5717400633440 T^{4} + 6124014171571617408 T^{6} + \)\(46\!\cdots\!31\)\( T^{8} + \)\(26\!\cdots\!72\)\( T^{10} + \)\(22\!\cdots\!00\)\( p^{2} T^{12} + \)\(26\!\cdots\!72\)\( p^{6} T^{14} + \)\(46\!\cdots\!31\)\( p^{12} T^{16} + 6124014171571617408 p^{18} T^{18} + 5717400633440 p^{24} T^{20} + 3427968 p^{30} T^{22} + p^{36} T^{24} \) | |
| 79 | \( 1 + 4422456 T^{2} + 9380100366532 T^{4} + 12679655224657630104 T^{6} + \)\(12\!\cdots\!87\)\( T^{8} + \)\(88\!\cdots\!96\)\( T^{10} + \)\(49\!\cdots\!40\)\( T^{12} + \)\(88\!\cdots\!96\)\( p^{6} T^{14} + \)\(12\!\cdots\!87\)\( p^{12} T^{16} + 12679655224657630104 p^{18} T^{18} + 9380100366532 p^{24} T^{20} + 4422456 p^{30} T^{22} + p^{36} T^{24} \) | |
| 83 | \( ( 1 + 1480 T + 3928102 T^{2} + 4094762656 T^{3} + 5957212821315 T^{4} + 4596756696874440 T^{5} + 4618007956869893676 T^{6} + 4596756696874440 p^{3} T^{7} + 5957212821315 p^{6} T^{8} + 4094762656 p^{9} T^{9} + 3928102 p^{12} T^{10} + 1480 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 89 | \( ( 1 + 1072 T + 3434266 T^{2} + 2533494768 T^{3} + 4909227644691 T^{4} + 2753123532874912 T^{5} + 4223825487645338740 T^{6} + 2753123532874912 p^{3} T^{7} + 4909227644691 p^{6} T^{8} + 2533494768 p^{9} T^{9} + 3434266 p^{12} T^{10} + 1072 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 97 | \( 1 + 5593248 T^{2} + 16968694787264 T^{4} + 35463927133823350432 T^{6} + \)\(56\!\cdots\!07\)\( T^{8} + \)\(70\!\cdots\!36\)\( T^{10} + \)\(71\!\cdots\!92\)\( T^{12} + \)\(70\!\cdots\!36\)\( p^{6} T^{14} + \)\(56\!\cdots\!07\)\( p^{12} T^{16} + 35463927133823350432 p^{18} T^{18} + 16968694787264 p^{24} T^{20} + 5593248 p^{30} T^{22} + p^{36} T^{24} \) | |
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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
−4.32927376484889706846712363410, −4.05426801494011383856624718381, −3.87838291463079950847578081580, −3.64662565561276960581137180689, −3.52843057022553843148639464275, −3.47986080197519134405289144752, −3.47836675348270245102783527031, −3.43103592106447743886612754842, −3.37053713058367719653000290860, −3.26309476058344674878165990297, −3.11079183814247211541913772885, −2.90195107330070120662980348654, −2.72257208992771570984289301690, −2.61661809863143040146612648359, −2.58662334850458093131607388487, −2.40543195498166811251681261515, −2.31997875171128384426861282065, −2.04184674760399630399440607845, −1.79327413601333073144987967281, −1.78644593411020734177600392640, −1.56615758347261809442965012993, −1.41697803216526841253758566484, −1.39811742401744527132762499171, −1.37992491709147743364544460760, −1.24481141146538019570351685906, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.24481141146538019570351685906, 1.37992491709147743364544460760, 1.39811742401744527132762499171, 1.41697803216526841253758566484, 1.56615758347261809442965012993, 1.78644593411020734177600392640, 1.79327413601333073144987967281, 2.04184674760399630399440607845, 2.31997875171128384426861282065, 2.40543195498166811251681261515, 2.58662334850458093131607388487, 2.61661809863143040146612648359, 2.72257208992771570984289301690, 2.90195107330070120662980348654, 3.11079183814247211541913772885, 3.26309476058344674878165990297, 3.37053713058367719653000290860, 3.43103592106447743886612754842, 3.47836675348270245102783527031, 3.47986080197519134405289144752, 3.52843057022553843148639464275, 3.64662565561276960581137180689, 3.87838291463079950847578081580, 4.05426801494011383856624718381, 4.32927376484889706846712363410
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.