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Properties

Label 8-31e8-1.1-c0e4-0-0
Degree 88
Conductor 852891037441852891037441
Sign 11
Analytic cond. 0.05290800.0529080
Root an. cond. 0.6925320.692532
Motivic weight 00
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  + 2-s + 4-s − 4·5-s + 7-s − 9-s − 4·10-s + 14-s − 18-s + 19-s − 4·20-s + 6·25-s + 28-s − 32-s − 4·35-s − 36-s + 38-s + 41-s + 4·45-s − 2·47-s + 49-s + 6·50-s + 59-s − 63-s − 64-s + 8·67-s − 4·70-s + 71-s + ⋯
L(s)  = 1  + 2-s + 4-s − 4·5-s + 7-s − 9-s − 4·10-s + 14-s − 18-s + 19-s − 4·20-s + 6·25-s + 28-s − 32-s − 4·35-s − 36-s + 38-s + 41-s + 4·45-s − 2·47-s + 49-s + 6·50-s + 59-s − 63-s − 64-s + 8·67-s − 4·70-s + 71-s + ⋯

Functional equation

Λ(s)=((318)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((318)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 31831^{8}
Sign: 11
Analytic conductor: 0.05290800.0529080
Root analytic conductor: 0.6925320.692532
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 318, ( :0,0,0,0), 1)(8,\ 31^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.63890263990.6389026399
L(12)L(\frac12) ≈\approx 0.63890263990.6389026399
L(1)L(1) 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)
bad31 1 1
good2C4×C2C_4\times C_2 1−T+T3−T4+T5−T7+T8 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8}
3C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
5C2C_2 (1+T+T2)4 ( 1 + T + T^{2} )^{4}
7C4×C2C_4\times C_2 1−T+T3−T4+T5−T7+T8 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8}
11C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
13C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
17C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
19C4×C2C_4\times C_2 1−T+T3−T4+T5−T7+T8 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8}
23C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
29C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
37C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
41C4×C2C_4\times C_2 1−T+T3−T4+T5−T7+T8 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8}
43C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
47C4C_4 (1+T+T2+T3+T4)2 ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
53C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
59C4×C2C_4\times C_2 1−T+T3−T4+T5−T7+T8 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8}
61C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
67C1C_1 (1−T)8 ( 1 - T )^{8}
71C4×C2C_4\times C_2 1−T+T3−T4+T5−T7+T8 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8}
73C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
79C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
83C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
89C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
97C4×C2C_4\times C_2 1−T+T3−T4+T5−T7+T8 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + 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

−7.39745140265450717735596783736, −7.36202380002155801657416394872, −7.03021885067196271822906544245, −6.74687668537547120398363177976, −6.58543501692979862242916410041, −6.39132316265927062708174563964, −5.90119211391729129107191851526, −5.61640950188835646776086368368, −5.58962474294699969164112923343, −5.38459402420678534271156286873, −4.92070586494791923869689276324, −4.74563035446808683017939455733, −4.70309477190647836673756099552, −4.15823935514117878179352947406, −3.99992463075727683850497885626, −3.91402161414175759583179288958, −3.41600072498515458727613160522, −3.40132014597364513401999463594, −3.38262884136449105992825694290, −3.07408450536182380060690743409, −2.22380342541582270248557549973, −2.18994136627161153429654453173, −2.12279685446713081119735335241, −0.990534801976033516665361488151, −0.67741350491287073570256142602, 0.67741350491287073570256142602, 0.990534801976033516665361488151, 2.12279685446713081119735335241, 2.18994136627161153429654453173, 2.22380342541582270248557549973, 3.07408450536182380060690743409, 3.38262884136449105992825694290, 3.40132014597364513401999463594, 3.41600072498515458727613160522, 3.91402161414175759583179288958, 3.99992463075727683850497885626, 4.15823935514117878179352947406, 4.70309477190647836673756099552, 4.74563035446808683017939455733, 4.92070586494791923869689276324, 5.38459402420678534271156286873, 5.58962474294699969164112923343, 5.61640950188835646776086368368, 5.90119211391729129107191851526, 6.39132316265927062708174563964, 6.58543501692979862242916410041, 6.74687668537547120398363177976, 7.03021885067196271822906544245, 7.36202380002155801657416394872, 7.39745140265450717735596783736

Graph of the ZZ-function along the critical line