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Properties

Label 8-308e4-1.1-c0e4-0-0
Degree 88
Conductor 89991784968999178496
Sign 11
Analytic cond. 0.0005582530.000558253
Root an. cond. 0.3920610.392061
Motivic weight 00
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-s − 7-s + 9-s − 11-s − 14-s + 18-s − 22-s − 25-s − 32-s − 3·37-s + 2·43-s − 50-s − 3·53-s − 63-s − 64-s + 5·71-s − 3·74-s + 77-s − 3·79-s + 2·86-s − 99-s − 3·106-s + 3·107-s − 2·113-s − 126-s + 127-s + 131-s + ⋯
L(s)  = 1  + 2-s − 7-s + 9-s − 11-s − 14-s + 18-s − 22-s − 25-s − 32-s − 3·37-s + 2·43-s − 50-s − 3·53-s − 63-s − 64-s + 5·71-s − 3·74-s + 77-s − 3·79-s + 2·86-s − 99-s − 3·106-s + 3·107-s − 2·113-s − 126-s + 127-s + 131-s + ⋯

Functional equation

Λ(s)=((28⋅74⋅114)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((28⋅74⋅114)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28⋅74⋅1142^{8} \cdot 7^{4} \cdot 11^{4}
Sign: 11
Analytic conductor: 0.0005582530.000558253
Root analytic conductor: 0.3920610.392061
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅74⋅114, ( :0,0,0,0), 1)(8,\ 2^{8} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.50008219960.5000821996
L(12)L(\frac12) ≈\approx 0.50008219960.5000821996
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)
bad2C4C_4 1−T+T2−T3+T4 1 - T + T^{2} - T^{3} + T^{4}
7C4C_4 1+T+T2+T3+T4 1 + T + T^{2} + T^{3} + T^{4}
11C4C_4 1+T+T2+T3+T4 1 + T + T^{2} + T^{3} + T^{4}
good3C4×C2C_4\times C_2 1−T2+T4−T6+T8 1 - T^{2} + T^{4} - T^{6} + T^{8}
5C4C_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} )
13C4×C2C_4\times C_2 1−T2+T4−T6+T8 1 - T^{2} + T^{4} - T^{6} + T^{8}
17C4×C2C_4\times C_2 1−T2+T4−T6+T8 1 - T^{2} + T^{4} - T^{6} + T^{8}
19C4C_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} )
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} )
31C4×C2C_4\times C_2 1−T2+T4−T6+T8 1 - T^{2} + T^{4} - T^{6} + T^{8}
37C1C_1×\timesC4C_4 (1+T)4(1−T+T2−T3+T4) ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )
41C4×C2C_4\times C_2 1−T2+T4−T6+T8 1 - T^{2} + T^{4} - T^{6} + T^{8}
43C4C_4 (1−T+T2−T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}
47C4×C2C_4\times C_2 1−T2+T4−T6+T8 1 - T^{2} + T^{4} - T^{6} + T^{8}
53C1C_1×\timesC4C_4 (1+T)4(1−T+T2−T3+T4) ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )
59C4×C2C_4\times C_2 1−T2+T4−T6+T8 1 - T^{2} + T^{4} - T^{6} + T^{8}
61C4×C2C_4\times C_2 1−T2+T4−T6+T8 1 - T^{2} + T^{4} - T^{6} + T^{8}
67C4C_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} )
71C1C_1×\timesC4C_4 (1−T)4(1−T+T2−T3+T4) ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )
73C4×C2C_4\times C_2 1−T2+T4−T6+T8 1 - T^{2} + T^{4} - T^{6} + T^{8}
79C1C_1×\timesC4C_4 (1+T)4(1−T+T2−T3+T4) ( 1 + 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} )
89C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
97C4C_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} )
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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

−8.881399509964142578173480154335, −8.404162664876180116596919548400, −8.229789242283892413502559059665, −7.86797693417786416749683553178, −7.74008318749546535126414633261, −7.44695276480689231780791241560, −7.25577357353975236947200284418, −6.77248739965654946590809032076, −6.73663614361854083901690259650, −6.35566491626277483957974503507, −6.30963892285751417809317182030, −5.64883353293780937296635881754, −5.48816545679951374629477683142, −5.22452612829304758347818127061, −5.21194012104362180367471639432, −4.53111078957030650189012729291, −4.43424937145661039226077371555, −4.20732118965755099865304951168, −3.73957420713434814425022933589, −3.45067266594494925634135014028, −3.26922858556763795880648304961, −2.86573205785808218237549130098, −2.20078112360369189262110859727, −2.01470570013651892664210429723, −1.41049860462560735985848609586, 1.41049860462560735985848609586, 2.01470570013651892664210429723, 2.20078112360369189262110859727, 2.86573205785808218237549130098, 3.26922858556763795880648304961, 3.45067266594494925634135014028, 3.73957420713434814425022933589, 4.20732118965755099865304951168, 4.43424937145661039226077371555, 4.53111078957030650189012729291, 5.21194012104362180367471639432, 5.22452612829304758347818127061, 5.48816545679951374629477683142, 5.64883353293780937296635881754, 6.30963892285751417809317182030, 6.35566491626277483957974503507, 6.73663614361854083901690259650, 6.77248739965654946590809032076, 7.25577357353975236947200284418, 7.44695276480689231780791241560, 7.74008318749546535126414633261, 7.86797693417786416749683553178, 8.229789242283892413502559059665, 8.404162664876180116596919548400, 8.881399509964142578173480154335

Graph of the ZZ-function along the critical line