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Properties

Label 24-338e12-1.1-c3e12-0-2
Degree 2424
Conductor 2.223×10302.223\times 10^{30}
Sign 11
Analytic cond. 3.95724×10153.95724\times 10^{15}
Root an. cond. 4.465714.46571
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 18·3-s − 24·4-s + 113·9-s − 432·12-s + 336·16-s − 198·17-s − 534·23-s + 382·25-s + 374·27-s − 238·29-s − 2.71e3·36-s − 856·43-s + 6.04e3·48-s + 1.15e3·49-s − 3.56e3·51-s + 178·53-s + 3.40e3·61-s − 3.58e3·64-s + 4.75e3·68-s − 9.61e3·69-s + 6.87e3·75-s − 1.75e3·79-s + 2.89e3·81-s − 4.28e3·87-s + 1.28e4·92-s − 9.16e3·100-s + 5.49e3·101-s + ⋯
L(s)  = 1  + 3.46·3-s − 3·4-s + 4.18·9-s − 10.3·12-s + 21/4·16-s − 2.82·17-s − 4.84·23-s + 3.05·25-s + 2.66·27-s − 1.52·29-s − 12.5·36-s − 3.03·43-s + 18.1·48-s + 3.37·49-s − 9.78·51-s + 0.461·53-s + 7.15·61-s − 7·64-s + 8.47·68-s − 16.7·69-s + 10.5·75-s − 2.49·79-s + 3.96·81-s − 5.27·87-s + 14.5·92-s − 9.16·100-s + 5.41·101-s + ⋯

Functional equation

Λ(s)=((212⋅1324)s/2 ΓC(s)12 L(s)=( Λ(4−s)\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}
Λ(s)=((212⋅1324)s/2 ΓC(s+3/2)12 L(s)=( Λ(1−s)\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: 2424
Conductor: 212⋅13242^{12} \cdot 13^{24}
Sign: 11
Analytic conductor: 3.95724×10153.95724\times 10^{15}
Root analytic conductor: 4.465714.46571
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 212⋅1324, ( :[3/2]12), 1)(24,\ 2^{12} \cdot 13^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.19730989380.1973098938
L(12)L(\frac12) ≈\approx 0.19730989380.1973098938
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 (1+p2T2)6 ( 1 + p^{2} T^{2} )^{6}
13 1 1
good3 (1−p2T+65T2−484T3+308p2T4−14446T5+82285T6−14446p3T7+308p8T8−484p9T9+65p12T10−p17T11+p18T12)2 ( 1 - p^{2} T + 65 T^{2} - 484 T^{3} + 308 p^{2} T^{4} - 14446 T^{5} + 82285 T^{6} - 14446 p^{3} T^{7} + 308 p^{8} T^{8} - 484 p^{9} T^{9} + 65 p^{12} T^{10} - p^{17} T^{11} + p^{18} T^{12} )^{2}
5 1−382T2+93199T4−3539669pT6+2754615289T8−389109534481T10+50512702160206T12−389109534481p6T14+2754615289p12T16−3539669p19T18+93199p24T20−382p30T22+p36T24 1 - 382 T^{2} + 93199 T^{4} - 3539669 p T^{6} + 2754615289 T^{8} - 389109534481 T^{10} + 50512702160206 T^{12} - 389109534481 p^{6} T^{14} + 2754615289 p^{12} T^{16} - 3539669 p^{19} T^{18} + 93199 p^{24} T^{20} - 382 p^{30} T^{22} + p^{36} T^{24}
7 1−1159T2+813634T4−415626322T6+25311558955pT8−1406264176663p2T10+496947802324576p2T12−1406264176663p8T14+25311558955p13T16−415626322p18T18+813634p24T20−1159p30T22+p36T24 1 - 1159 T^{2} + 813634 T^{4} - 415626322 T^{6} + 25311558955 p T^{8} - 1406264176663 p^{2} T^{10} + 496947802324576 p^{2} T^{12} - 1406264176663 p^{8} T^{14} + 25311558955 p^{13} T^{16} - 415626322 p^{18} T^{18} + 813634 p^{24} T^{20} - 1159 p^{30} T^{22} + p^{36} T^{24}
11 1−8117T2+33989553T4−98393356134T6+219078606548874T8−391842639166411830T10+ 1 - 8117 T^{2} + 33989553 T^{4} - 98393356134 T^{6} + 219078606548874 T^{8} - 391842639166411830 T^{10} + 57 ⁣⋯ ⁣3557\!\cdots\!35T12−391842639166411830p6T14+219078606548874p12T16−98393356134p18T18+33989553p24T20−8117p30T22+p36T24 T^{12} - 391842639166411830 p^{6} T^{14} + 219078606548874 p^{12} T^{16} - 98393356134 p^{18} T^{18} + 33989553 p^{24} T^{20} - 8117 p^{30} T^{22} + p^{36} T^{24}
17 (1+99T+15235T2+1418010T3+155831450T4+11553597880T5+895771727153T6+11553597880p3T7+155831450p6T8+1418010p9T9+15235p12T10+99p15T11+p18T12)2 ( 1 + 99 T + 15235 T^{2} + 1418010 T^{3} + 155831450 T^{4} + 11553597880 T^{5} + 895771727153 T^{6} + 11553597880 p^{3} T^{7} + 155831450 p^{6} T^{8} + 1418010 p^{9} T^{9} + 15235 p^{12} T^{10} + 99 p^{15} T^{11} + p^{18} T^{12} )^{2}
19 1−36013T2+716146137T4−10102929361070T6+111338486468995570T8− 1 - 36013 T^{2} + 716146137 T^{4} - 10102929361070 T^{6} + 111338486468995570 T^{8} - 10 ⁣⋯ ⁣1410\!\cdots\!14T10+ T^{10} + 74 ⁣⋯ ⁣3974\!\cdots\!39T12− T^{12} - 10 ⁣⋯ ⁣1410\!\cdots\!14p6T14+111338486468995570p12T16−10102929361070p18T18+716146137p24T20−36013p30T22+p36T24 p^{6} T^{14} + 111338486468995570 p^{12} T^{16} - 10102929361070 p^{18} T^{18} + 716146137 p^{24} T^{20} - 36013 p^{30} T^{22} + p^{36} T^{24}
23 (1+267T+65293T2+11352002T3+1904634711T4+10605914921pT5+29697637919118T6+10605914921p4T7+1904634711p6T8+11352002p9T9+65293p12T10+267p15T11+p18T12)2 ( 1 + 267 T + 65293 T^{2} + 11352002 T^{3} + 1904634711 T^{4} + 10605914921 p T^{5} + 29697637919118 T^{6} + 10605914921 p^{4} T^{7} + 1904634711 p^{6} T^{8} + 11352002 p^{9} T^{9} + 65293 p^{12} T^{10} + 267 p^{15} T^{11} + p^{18} T^{12} )^{2}
29 (1+119T+124344T2+11836678T3+6759160701T4+516098000019T5+7271394875556pT6+516098000019p3T7+6759160701p6T8+11836678p9T9+124344p12T10+119p15T11+p18T12)2 ( 1 + 119 T + 124344 T^{2} + 11836678 T^{3} + 6759160701 T^{4} + 516098000019 T^{5} + 7271394875556 p T^{6} + 516098000019 p^{3} T^{7} + 6759160701 p^{6} T^{8} + 11836678 p^{9} T^{9} + 124344 p^{12} T^{10} + 119 p^{15} T^{11} + p^{18} T^{12} )^{2}
31 1−241655T2+28942573242T4−2264770361862706T6+ 1 - 241655 T^{2} + 28942573242 T^{4} - 2264770361862706 T^{6} + 12 ⁣⋯ ⁣5712\!\cdots\!57T8− T^{8} - 55 ⁣⋯ ⁣0755\!\cdots\!07T10+ T^{10} + 18 ⁣⋯ ⁣2018\!\cdots\!20T12− T^{12} - 55 ⁣⋯ ⁣0755\!\cdots\!07p6T14+ p^{6} T^{14} + 12 ⁣⋯ ⁣5712\!\cdots\!57p12T16−2264770361862706p18T18+28942573242p24T20−241655p30T22+p36T24 p^{12} T^{16} - 2264770361862706 p^{18} T^{18} + 28942573242 p^{24} T^{20} - 241655 p^{30} T^{22} + p^{36} T^{24}
37 1−419434T2+87929952639T4−11996835934207205T6+ 1 - 419434 T^{2} + 87929952639 T^{4} - 11996835934207205 T^{6} + 11 ⁣⋯ ⁣3311\!\cdots\!33T8− T^{8} - 87 ⁣⋯ ⁣6187\!\cdots\!61T10+ T^{10} + 50 ⁣⋯ ⁣8650\!\cdots\!86T12− T^{12} - 87 ⁣⋯ ⁣6187\!\cdots\!61p6T14+ p^{6} T^{14} + 11 ⁣⋯ ⁣3311\!\cdots\!33p12T16−11996835934207205p18T18+87929952639p24T20−419434p30T22+p36T24 p^{12} T^{16} - 11996835934207205 p^{18} T^{18} + 87929952639 p^{24} T^{20} - 419434 p^{30} T^{22} + p^{36} T^{24}
41 1−435796T2+88505255842T4−11683055467962385T6+ 1 - 435796 T^{2} + 88505255842 T^{4} - 11683055467962385 T^{6} + 12 ⁣⋯ ⁣4712\!\cdots\!47T8− T^{8} - 10 ⁣⋯ ⁣9410\!\cdots\!94T10+ T^{10} + 79 ⁣⋯ ⁣9179\!\cdots\!91T12− T^{12} - 10 ⁣⋯ ⁣9410\!\cdots\!94p6T14+ p^{6} T^{14} + 12 ⁣⋯ ⁣4712\!\cdots\!47p12T16−11683055467962385p18T18+88505255842p24T20−435796p30T22+p36T24 p^{12} T^{16} - 11683055467962385 p^{18} T^{18} + 88505255842 p^{24} T^{20} - 435796 p^{30} T^{22} + p^{36} T^{24}
43 (1+428T+199837T2+75716384T3+36167871145T4+9851740849988T5+2980287990273901T6+9851740849988p3T7+36167871145p6T8+75716384p9T9+199837p12T10+428p15T11+p18T12)2 ( 1 + 428 T + 199837 T^{2} + 75716384 T^{3} + 36167871145 T^{4} + 9851740849988 T^{5} + 2980287990273901 T^{6} + 9851740849988 p^{3} T^{7} + 36167871145 p^{6} T^{8} + 75716384 p^{9} T^{9} + 199837 p^{12} T^{10} + 428 p^{15} T^{11} + p^{18} T^{12} )^{2}
47 1−453138T2+135321770311T4−29047321613511285T6+ 1 - 453138 T^{2} + 135321770311 T^{4} - 29047321613511285 T^{6} + 49 ⁣⋯ ⁣4149\!\cdots\!41T8− T^{8} - 67 ⁣⋯ ⁣2167\!\cdots\!21T10+ T^{10} + 76 ⁣⋯ ⁣0276\!\cdots\!02T12− T^{12} - 67 ⁣⋯ ⁣2167\!\cdots\!21p6T14+ p^{6} T^{14} + 49 ⁣⋯ ⁣4149\!\cdots\!41p12T16−29047321613511285p18T18+135321770311p24T20−453138p30T22+p36T24 p^{12} T^{16} - 29047321613511285 p^{18} T^{18} + 135321770311 p^{24} T^{20} - 453138 p^{30} T^{22} + p^{36} T^{24}
53 (1−89T+149636T2+19578912T3+11883866543T4−8498293720903T5+5004078025030288T6−8498293720903p3T7+11883866543p6T8+19578912p9T9+149636p12T10−89p15T11+p18T12)2 ( 1 - 89 T + 149636 T^{2} + 19578912 T^{3} + 11883866543 T^{4} - 8498293720903 T^{5} + 5004078025030288 T^{6} - 8498293720903 p^{3} T^{7} + 11883866543 p^{6} T^{8} + 19578912 p^{9} T^{9} + 149636 p^{12} T^{10} - 89 p^{15} T^{11} + p^{18} T^{12} )^{2}
59 1−526184T2+141586970478T4−41662826122817849T6+ 1 - 526184 T^{2} + 141586970478 T^{4} - 41662826122817849 T^{6} + 13 ⁣⋯ ⁣7113\!\cdots\!71T8− T^{8} - 29 ⁣⋯ ⁣5429\!\cdots\!54T10+ T^{10} + 57 ⁣⋯ ⁣1557\!\cdots\!15T12− T^{12} - 29 ⁣⋯ ⁣5429\!\cdots\!54p6T14+ p^{6} T^{14} + 13 ⁣⋯ ⁣7113\!\cdots\!71p12T16−41662826122817849p18T18+141586970478p24T20−526184p30T22+p36T24 p^{12} T^{16} - 41662826122817849 p^{18} T^{18} + 141586970478 p^{24} T^{20} - 526184 p^{30} T^{22} + p^{36} T^{24}
61 (1−1704T+1938703T2−1635367461T3+1162035618775T4−684320852667059T5+351976137165513138T6−684320852667059p3T7+1162035618775p6T8−1635367461p9T9+1938703p12T10−1704p15T11+p18T12)2 ( 1 - 1704 T + 1938703 T^{2} - 1635367461 T^{3} + 1162035618775 T^{4} - 684320852667059 T^{5} + 351976137165513138 T^{6} - 684320852667059 p^{3} T^{7} + 1162035618775 p^{6} T^{8} - 1635367461 p^{9} T^{9} + 1938703 p^{12} T^{10} - 1704 p^{15} T^{11} + p^{18} T^{12} )^{2}
67 1−2608162T2+3205674513359T4−2491312924906790680T6+ 1 - 2608162 T^{2} + 3205674513359 T^{4} - 2491312924906790680 T^{6} + 13 ⁣⋯ ⁣2113\!\cdots\!21T8− T^{8} - 58 ⁣⋯ ⁣7458\!\cdots\!74T10+ T^{10} + 19 ⁣⋯ ⁣3119\!\cdots\!31T12− T^{12} - 58 ⁣⋯ ⁣7458\!\cdots\!74p6T14+ p^{6} T^{14} + 13 ⁣⋯ ⁣2113\!\cdots\!21p12T16−2491312924906790680p18T18+3205674513359p24T20−2608162p30T22+p36T24 p^{12} T^{16} - 2491312924906790680 p^{18} T^{18} + 3205674513359 p^{24} T^{20} - 2608162 p^{30} T^{22} + p^{36} T^{24}
71 1−1935299T2+2038830960730T4−1525475006651412638T6+ 1 - 1935299 T^{2} + 2038830960730 T^{4} - 1525475006651412638 T^{6} + 89 ⁣⋯ ⁣4989\!\cdots\!49T8− T^{8} - 42 ⁣⋯ ⁣9142\!\cdots\!91T10+ T^{10} + 16 ⁣⋯ ⁣9216\!\cdots\!92T12− T^{12} - 42 ⁣⋯ ⁣9142\!\cdots\!91p6T14+ p^{6} T^{14} + 89 ⁣⋯ ⁣4989\!\cdots\!49p12T16−1525475006651412638p18T18+2038830960730p24T20−1935299p30T22+p36T24 p^{12} T^{16} - 1525475006651412638 p^{18} T^{18} + 2038830960730 p^{24} T^{20} - 1935299 p^{30} T^{22} + p^{36} T^{24}
73 1−2935638T2+3898939340527T4−3093494767983198216T6+ 1 - 2935638 T^{2} + 3898939340527 T^{4} - 3093494767983198216 T^{6} + 16 ⁣⋯ ⁣6116\!\cdots\!61T8− T^{8} - 68 ⁣⋯ ⁣8268\!\cdots\!82T10+ T^{10} + 25 ⁣⋯ ⁣3525\!\cdots\!35T12− T^{12} - 68 ⁣⋯ ⁣8268\!\cdots\!82p6T14+ p^{6} T^{14} + 16 ⁣⋯ ⁣6116\!\cdots\!61p12T16−3093494767983198216p18T18+3898939340527p24T20−2935638p30T22+p36T24 p^{12} T^{16} - 3093494767983198216 p^{18} T^{18} + 3898939340527 p^{24} T^{20} - 2935638 p^{30} T^{22} + p^{36} T^{24}
79 (1+875T+2505744T2+1612380868T3+2680533156433T4+1358001363523637T5+1672684167171147060T6+1358001363523637p3T7+2680533156433p6T8+1612380868p9T9+2505744p12T10+875p15T11+p18T12)2 ( 1 + 875 T + 2505744 T^{2} + 1612380868 T^{3} + 2680533156433 T^{4} + 1358001363523637 T^{5} + 1672684167171147060 T^{6} + 1358001363523637 p^{3} T^{7} + 2680533156433 p^{6} T^{8} + 1612380868 p^{9} T^{9} + 2505744 p^{12} T^{10} + 875 p^{15} T^{11} + p^{18} T^{12} )^{2}
83 1−3141070T2+5512039858027T4−6761964563893293052T6+ 1 - 3141070 T^{2} + 5512039858027 T^{4} - 6761964563893293052 T^{6} + 64 ⁣⋯ ⁣4964\!\cdots\!49T8− T^{8} - 48 ⁣⋯ ⁣5448\!\cdots\!54T10+ T^{10} + 30 ⁣⋯ ⁣9530\!\cdots\!95T12− T^{12} - 48 ⁣⋯ ⁣5448\!\cdots\!54p6T14+ p^{6} T^{14} + 64 ⁣⋯ ⁣4964\!\cdots\!49p12T16−6761964563893293052p18T18+5512039858027p24T20−3141070p30T22+p36T24 p^{12} T^{16} - 6761964563893293052 p^{18} T^{18} + 5512039858027 p^{24} T^{20} - 3141070 p^{30} T^{22} + p^{36} T^{24}
89 1−4477394T2+9702807750839T4−157611520547438048pT6+ 1 - 4477394 T^{2} + 9702807750839 T^{4} - 157611520547438048 p T^{6} + 15 ⁣⋯ ⁣1315\!\cdots\!13T8− T^{8} - 14 ⁣⋯ ⁣5014\!\cdots\!50T10+ T^{10} + 11 ⁣⋯ ⁣0311\!\cdots\!03T12− T^{12} - 14 ⁣⋯ ⁣5014\!\cdots\!50p6T14+ p^{6} T^{14} + 15 ⁣⋯ ⁣1315\!\cdots\!13p12T16−157611520547438048p19T18+9702807750839p24T20−4477394p30T22+p36T24 p^{12} T^{16} - 157611520547438048 p^{19} T^{18} + 9702807750839 p^{24} T^{20} - 4477394 p^{30} T^{22} + p^{36} T^{24}
97 1−9021806T2+37946584520559T4−99120815383266274472T6+ 1 - 9021806 T^{2} + 37946584520559 T^{4} - 99120815383266274472 T^{6} + 18 ⁣⋯ ⁣0918\!\cdots\!09T8− T^{8} - 24 ⁣⋯ ⁣6624\!\cdots\!66T10+ T^{10} + 25 ⁣⋯ ⁣6325\!\cdots\!63T12− T^{12} - 24 ⁣⋯ ⁣6624\!\cdots\!66p6T14+ p^{6} T^{14} + 18 ⁣⋯ ⁣0918\!\cdots\!09p12T16−99120815383266274472p18T18+37946584520559p24T20−9021806p30T22+p36T24 p^{12} T^{16} - 99120815383266274472 p^{18} T^{18} + 37946584520559 p^{24} T^{20} - 9021806 p^{30} T^{22} + p^{36} T^{24}
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   L(s)=∏p ∏j=124(1−αj,p p−s)−1L(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

−3.57256431375112625004671972238, −3.19532046482596558335048873384, −3.09582884559210710658992061698, −3.00893219717186423791829373622, −2.97280603610110326183609912313, −2.75274305867495526505873189976, −2.69370145396242740628232150211, −2.64962447015437579581122543019, −2.48058402945167771498317618514, −2.42960978256932254725045077543, −2.26474035264718660882074867350, −2.15679130988256361183891440009, −1.98952499002397111435051654659, −1.87532970908653512016328609221, −1.81992222531118537882535967640, −1.79135427229283974094067989148, −1.18605597315673265785970175965, −1.12619515800567375010360624990, −1.07929394871725255604005261470, −1.02267305121868424474347496869, −0.965570939166470091265330792405, −0.44156958468892323436473299486, −0.25488733965988766285706209639, −0.23764842748265696170937952456, −0.04768520933474828476461476316, 0.04768520933474828476461476316, 0.23764842748265696170937952456, 0.25488733965988766285706209639, 0.44156958468892323436473299486, 0.965570939166470091265330792405, 1.02267305121868424474347496869, 1.07929394871725255604005261470, 1.12619515800567375010360624990, 1.18605597315673265785970175965, 1.79135427229283974094067989148, 1.81992222531118537882535967640, 1.87532970908653512016328609221, 1.98952499002397111435051654659, 2.15679130988256361183891440009, 2.26474035264718660882074867350, 2.42960978256932254725045077543, 2.48058402945167771498317618514, 2.64962447015437579581122543019, 2.69370145396242740628232150211, 2.75274305867495526505873189976, 2.97280603610110326183609912313, 3.00893219717186423791829373622, 3.09582884559210710658992061698, 3.19532046482596558335048873384, 3.57256431375112625004671972238

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.