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Properties

Label 8-832e4-1.1-c1e4-0-7
Degree 88
Conductor 479174066176479174066176
Sign 11
Analytic cond. 1948.051948.05
Root an. cond. 2.577502.57750
Motivic weight 11
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  − 6·3-s + 6·5-s + 8·7-s + 16·9-s − 12·11-s + 6·13-s − 36·15-s − 12·17-s + 2·19-s − 48·21-s − 6·23-s + 18·25-s − 24·27-s − 6·29-s + 8·31-s + 72·33-s + 48·35-s + 8·37-s − 36·39-s − 12·41-s + 4·43-s + 96·45-s + 24·47-s + 44·49-s + 72·51-s + 24·53-s − 72·55-s + ⋯
L(s)  = 1  − 3.46·3-s + 2.68·5-s + 3.02·7-s + 16/3·9-s − 3.61·11-s + 1.66·13-s − 9.29·15-s − 2.91·17-s + 0.458·19-s − 10.4·21-s − 1.25·23-s + 18/5·25-s − 4.61·27-s − 1.11·29-s + 1.43·31-s + 12.5·33-s + 8.11·35-s + 1.31·37-s − 5.76·39-s − 1.87·41-s + 0.609·43-s + 14.3·45-s + 3.50·47-s + 44/7·49-s + 10.0·51-s + 3.29·53-s − 9.70·55-s + ⋯

Functional equation

Λ(s)=((224⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅1342^{24} \cdot 13^{4}
Sign: 11
Analytic conductor: 1948.051948.05
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.1525647071.152564707
L(12)L(\frac12) ≈\approx 1.1525647071.152564707
L(32)L(\frac{3}{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)
bad2 1 1
13C22C_2^2 1−6T+23T2−6pT3+p2T4 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
good3D4×C2D_4\times C_2 1+2pT+20T2+16pT3+91T4+16p2T5+20p2T6+2p4T7+p4T8 1 + 2 p T + 20 T^{2} + 16 p T^{3} + 91 T^{4} + 16 p^{2} T^{5} + 20 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8}
5C22C_2^2×\timesC22C_2^2 (1−4T+11T2−4pT3+p2T4)(1−2T−T2−2pT3+p2T4) ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )
7D4×C2D_4\times C_2 1−8T+20T2+12T3−145T4+12pT5+20p2T6−8p3T7+p4T8 1 - 8 T + 20 T^{2} + 12 T^{3} - 145 T^{4} + 12 p T^{5} + 20 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
11C23C_2^3 1+12T+72T2+288T3+983T4+288pT5+72p2T6+12p3T7+p4T8 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 288 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
17D4×C2D_4\times C_2 1+12T+5pT2+444T3+1896T4+444pT5+5p3T6+12p3T7+p4T8 1 + 12 T + 5 p T^{2} + 444 T^{3} + 1896 T^{4} + 444 p T^{5} + 5 p^{3} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
19C2C_2×\timesC22C_2^2 (1−T+pT2)2(1+11T2+p2T4) ( 1 - T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} )
23D4×C2D_4\times C_2 1+6T−16T2+36T3+1347T4+36pT5−16p2T6+6p3T7+p4T8 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
29C22C_2^2 (1+3T−20T2+3pT3+p2T4)2 ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}
31D4×C2D_4\times C_2 1−8T+32T2−264T3+2174T4−264pT5+32p2T6−8p3T7+p4T8 1 - 8 T + 32 T^{2} - 264 T^{3} + 2174 T^{4} - 264 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
37C2C_2×\timesC22C_2^2 (1+2T+pT2)2(1−12T+107T2−12pT3+p2T4) ( 1 + 2 T + p T^{2} )^{2}( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )
41D4×C2D_4\times C_2 1+12T+117T2+840T3+5804T4+840pT5+117p2T6+12p3T7+p4T8 1 + 12 T + 117 T^{2} + 840 T^{3} + 5804 T^{4} + 840 p T^{5} + 117 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
43C22C_2^2 (1−2T−39T2−2pT3+p2T4)2 ( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
47D4×C2D_4\times C_2 1−24T+288T2−2712T3+21182T4−2712pT5+288p2T6−24p3T7+p4T8 1 - 24 T + 288 T^{2} - 2712 T^{3} + 21182 T^{4} - 2712 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}
53D4D_{4} (1−12T+139T2−12pT3+p2T4)2 ( 1 - 12 T + 139 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1+12T+36T2−12pT3−9001T4−12p2T5+36p2T6+12p3T7+p4T8 1 + 12 T + 36 T^{2} - 12 p T^{3} - 9001 T^{4} - 12 p^{2} T^{5} + 36 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
61C23C_2^3 1−95T2+5304T4−95p2T6+p4T8 1 - 95 T^{2} + 5304 T^{4} - 95 p^{2} T^{6} + p^{4} T^{8}
67D4×C2D_4\times C_2 1−10T+194T2−1644T3+18239T4−1644pT5+194p2T6−10p3T7+p4T8 1 - 10 T + 194 T^{2} - 1644 T^{3} + 18239 T^{4} - 1644 p T^{5} + 194 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}
71C23C_2^3 1−6T+18T2−36T3−4153T4−36pT5+18p2T6−6p3T7+p4T8 1 - 6 T + 18 T^{2} - 36 T^{3} - 4153 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
73D4×C2D_4\times C_2 1+14T+98T2+1176T3+13991T4+1176pT5+98p2T6+14p3T7+p4T8 1 + 14 T + 98 T^{2} + 1176 T^{3} + 13991 T^{4} + 1176 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}
79D4×C2D_4\times C_2 1−92T2+12870T4−92p2T6+p4T8 1 - 92 T^{2} + 12870 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8}
83D4×C2D_4\times C_2 1+12T+72T2+564T3+3122T4+564pT5+72p2T6+12p3T7+p4T8 1 + 12 T + 72 T^{2} + 564 T^{3} + 3122 T^{4} + 564 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
89C23C_2^3 1−18T+162T2−972T3+4607T4−972pT5+162p2T6−18p3T7+p4T8 1 - 18 T + 162 T^{2} - 972 T^{3} + 4607 T^{4} - 972 p T^{5} + 162 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}
97D4×C2D_4\times C_2 1−26T+2pT2+1332T3−32593T4+1332pT5+2p3T6−26p3T7+p4T8 1 - 26 T + 2 p T^{2} + 1332 T^{3} - 32593 T^{4} + 1332 p T^{5} + 2 p^{3} T^{6} - 26 p^{3} T^{7} + p^{4} 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

−7.36039100459451827030183113297, −7.01114795513029696137610828163, −6.74649012842567371443288662366, −6.28792554110948242848040496863, −6.09305563543480897186860096597, −6.00650846145920164572244645303, −5.93562871194940635801598101801, −5.75974631652799478148637042073, −5.50973482323966908056885013825, −5.32250500447841829932095173491, −4.99879187951165816534608027134, −4.96866723654543807378392220498, −4.86405898065164734075100601748, −4.54107629869485486917867626128, −4.21922874103801157219867621943, −3.95153074090327286059396727939, −3.53436304763043971305971134997, −2.56296925648340434243643081461, −2.49311733868751128281371914291, −2.39191378306723175061536677328, −1.98993652277652155637907004600, −1.94830438493260294608273274510, −1.20535291398892866975302546881, −0.906688163752870335751279372566, −0.40660087288661687601838351112, 0.40660087288661687601838351112, 0.906688163752870335751279372566, 1.20535291398892866975302546881, 1.94830438493260294608273274510, 1.98993652277652155637907004600, 2.39191378306723175061536677328, 2.49311733868751128281371914291, 2.56296925648340434243643081461, 3.53436304763043971305971134997, 3.95153074090327286059396727939, 4.21922874103801157219867621943, 4.54107629869485486917867626128, 4.86405898065164734075100601748, 4.96866723654543807378392220498, 4.99879187951165816534608027134, 5.32250500447841829932095173491, 5.50973482323966908056885013825, 5.75974631652799478148637042073, 5.93562871194940635801598101801, 6.00650846145920164572244645303, 6.09305563543480897186860096597, 6.28792554110948242848040496863, 6.74649012842567371443288662366, 7.01114795513029696137610828163, 7.36039100459451827030183113297

Graph of the ZZ-function along the critical line