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Properties

Label 8-525e4-1.1-c2e4-0-5
Degree 88
Conductor 7596914062575969140625
Sign 11
Analytic cond. 41877.141877.1
Root an. cond. 3.782223.78222
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 6·3-s + 3·4-s − 12·6-s + 26·7-s − 6·8-s + 21·9-s + 4·11-s − 18·12-s + 52·14-s − 12·16-s − 36·17-s + 42·18-s + 30·19-s − 156·21-s + 8·22-s − 40·23-s + 36·24-s − 54·27-s + 78·28-s − 8·29-s − 42·32-s − 24·33-s − 72·34-s + 63·36-s − 50·37-s + 60·38-s + ⋯
L(s)  = 1  + 2-s − 2·3-s + 3/4·4-s − 2·6-s + 26/7·7-s − 3/4·8-s + 7/3·9-s + 4/11·11-s − 3/2·12-s + 26/7·14-s − 3/4·16-s − 2.11·17-s + 7/3·18-s + 1.57·19-s − 7.42·21-s + 4/11·22-s − 1.73·23-s + 3/2·24-s − 2·27-s + 2.78·28-s − 0.275·29-s − 1.31·32-s − 0.727·33-s − 2.11·34-s + 7/4·36-s − 1.35·37-s + 1.57·38-s + ⋯

Functional equation

Λ(s)=((34⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((34⋅58⋅74)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅58⋅743^{4} \cdot 5^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 41877.141877.1
Root analytic conductor: 3.782223.78222
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅58⋅74, ( :1,1,1,1), 1)(8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 4.8459300124.845930012
L(12)L(\frac12) ≈\approx 4.8459300124.845930012
L(2)L(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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3C2C_2 (1+pT+pT2)2 ( 1 + p T + p T^{2} )^{2}
5 1 1
7C2C_2 (1−13T+p2T2)2 ( 1 - 13 T + p^{2} T^{2} )^{2}
good2D4×C2D_4\times C_2 1−pT+T2+5pT3−23T4+5p3T5+p4T6−p7T7+p8T8 1 - p T + T^{2} + 5 p T^{3} - 23 T^{4} + 5 p^{3} T^{5} + p^{4} T^{6} - p^{7} T^{7} + p^{8} T^{8}
11D4×C2D_4\times C_2 1−4T−206T2+80T3+32707T4+80p2T5−206p4T6−4p6T7+p8T8 1 - 4 T - 206 T^{2} + 80 T^{3} + 32707 T^{4} + 80 p^{2} T^{5} - 206 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8}
13D4×C2D_4\times C_2 1−382T2+72003T4−382p4T6+p8T8 1 - 382 T^{2} + 72003 T^{4} - 382 p^{4} T^{6} + p^{8} T^{8}
17D4×C2D_4\times C_2 1+36T+1046T2+22104T3+418323T4+22104p2T5+1046p4T6+36p6T7+p8T8 1 + 36 T + 1046 T^{2} + 22104 T^{3} + 418323 T^{4} + 22104 p^{2} T^{5} + 1046 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8}
19D4×C2D_4\times C_2 1−30T+449T2−4470T3+180T4−4470p2T5+449p4T6−30p6T7+p8T8 1 - 30 T + 449 T^{2} - 4470 T^{3} + 180 T^{4} - 4470 p^{2} T^{5} + 449 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8}
23C22C_2^2 (1+20T−129T2+20p2T3+p4T4)2 ( 1 + 20 T - 129 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2}
29D4D_{4} (1+4T+822T2+4p2T3+p4T4)2 ( 1 + 4 T + 822 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2}
31C23C_2^3 1+770T2−330621T4+770p4T6+p8T8 1 + 770 T^{2} - 330621 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8}
37D4×C2D_4\times C_2 1+50T−263T2+1250T3+2337508T4+1250p2T5−263p4T6+50p6T7+p8T8 1 + 50 T - 263 T^{2} + 1250 T^{3} + 2337508 T^{4} + 1250 p^{2} T^{5} - 263 p^{4} T^{6} + 50 p^{6} T^{7} + p^{8} T^{8}
41D4×C2D_4\times C_2 1+932T2+1635078T4+932p4T6+p8T8 1 + 932 T^{2} + 1635078 T^{4} + 932 p^{4} T^{6} + p^{8} T^{8}
43D4D_{4} (1+100T+6102T2+100p2T3+p4T4)2 ( 1 + 100 T + 6102 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2}
47D4×C2D_4\times C_2 1−144T+12410T2−791712T3+40616931T4−791712p2T5+12410p4T6−144p6T7+p8T8 1 - 144 T + 12410 T^{2} - 791712 T^{3} + 40616931 T^{4} - 791712 p^{2} T^{5} + 12410 p^{4} T^{6} - 144 p^{6} T^{7} + p^{8} T^{8}
53D4×C2D_4\times C_2 1−44T+1234T2+216304T3−12835901T4+216304p2T5+1234p4T6−44p6T7+p8T8 1 - 44 T + 1234 T^{2} + 216304 T^{3} - 12835901 T^{4} + 216304 p^{2} T^{5} + 1234 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8}
59D4×C2D_4\times C_2 1+24T+1370T2+28272T3−10061325T4+28272p2T5+1370p4T6+24p6T7+p8T8 1 + 24 T + 1370 T^{2} + 28272 T^{3} - 10061325 T^{4} + 28272 p^{2} T^{5} + 1370 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8}
61D4×C2D_4\times C_2 1+234T+29609T2+2657538T3+183051300T4+2657538p2T5+29609p4T6+234p6T7+p8T8 1 + 234 T + 29609 T^{2} + 2657538 T^{3} + 183051300 T^{4} + 2657538 p^{2} T^{5} + 29609 p^{4} T^{6} + 234 p^{6} T^{7} + p^{8} T^{8}
67D4×C2D_4\times C_2 1+2T−311T2−17326T3−20069852T4−17326p2T5−311p4T6+2p6T7+p8T8 1 + 2 T - 311 T^{2} - 17326 T^{3} - 20069852 T^{4} - 17326 p^{2} T^{5} - 311 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8}
71D4D_{4} (1+52T+8358T2+52p2T3+p4T4)2 ( 1 + 52 T + 8358 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{2}
73D4×C2D_4\times C_2 1−90T+13745T2−994050T3+107982084T4−994050p2T5+13745p4T6−90p6T7+p8T8 1 - 90 T + 13745 T^{2} - 994050 T^{3} + 107982084 T^{4} - 994050 p^{2} T^{5} + 13745 p^{4} T^{6} - 90 p^{6} T^{7} + p^{8} T^{8}
79D4×C2D_4\times C_2 1−38T−9863T2+44650T3+79886164T4+44650p2T5−9863p4T6−38p6T7+p8T8 1 - 38 T - 9863 T^{2} + 44650 T^{3} + 79886164 T^{4} + 44650 p^{2} T^{5} - 9863 p^{4} T^{6} - 38 p^{6} T^{7} + p^{8} T^{8}
83D4×C2D_4\times C_2 1−20500T2+187537542T4−20500p4T6+p8T8 1 - 20500 T^{2} + 187537542 T^{4} - 20500 p^{4} T^{6} + p^{8} T^{8}
89D4×C2D_4\times C_2 1+72T+16202T2+1042128T3+160441923T4+1042128p2T5+16202p4T6+72p6T7+p8T8 1 + 72 T + 16202 T^{2} + 1042128 T^{3} + 160441923 T^{4} + 1042128 p^{2} T^{5} + 16202 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8}
97D4×C2D_4\times C_2 1−15166T2+198604227T4−15166p4T6+p8T8 1 - 15166 T^{2} + 198604227 T^{4} - 15166 p^{4} T^{6} + p^{8} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.48648631182100294558106381320, −7.47714660131268725102083522968, −7.21196020824932607001040810802, −6.63144955488260325116389368751, −6.53002075501113332818534225022, −6.50484221001224225464639250015, −5.88625612494286435374801272161, −5.63575755339058352695631919696, −5.60909758371248619408964207249, −5.57458008704128769616970542677, −5.14478201505659318763634002656, −4.76631040383731409946451209957, −4.67045841760122013299132733327, −4.52237972443779369621808524106, −4.28138655270854586376728366600, −4.03026283542381001134254350325, −3.39079579403300998126143041405, −3.31337112171294096463724065765, −2.91342508322134079630602762481, −1.99549176162360804685915768498, −1.97868151263975733608444574943, −1.76987334655824090688519683427, −1.59671249791477935225535253902, −0.62373707054128203683210487197, −0.53477630185195833696425007371, 0.53477630185195833696425007371, 0.62373707054128203683210487197, 1.59671249791477935225535253902, 1.76987334655824090688519683427, 1.97868151263975733608444574943, 1.99549176162360804685915768498, 2.91342508322134079630602762481, 3.31337112171294096463724065765, 3.39079579403300998126143041405, 4.03026283542381001134254350325, 4.28138655270854586376728366600, 4.52237972443779369621808524106, 4.67045841760122013299132733327, 4.76631040383731409946451209957, 5.14478201505659318763634002656, 5.57458008704128769616970542677, 5.60909758371248619408964207249, 5.63575755339058352695631919696, 5.88625612494286435374801272161, 6.50484221001224225464639250015, 6.53002075501113332818534225022, 6.63144955488260325116389368751, 7.21196020824932607001040810802, 7.47714660131268725102083522968, 7.48648631182100294558106381320

Graph of the ZZ-function along the critical line