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Properties

Label 8-95e4-1.1-c1e4-0-2
Degree 88
Conductor 8145062581450625
Sign 11
Analytic cond. 0.3311330.331133
Root an. cond. 0.8709640.870964
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  + 2·5-s − 6·9-s − 4·11-s + 4·16-s + 14·19-s + 5·25-s + 18·29-s − 28·31-s − 4·41-s − 12·45-s − 4·49-s − 8·55-s + 18·59-s + 14·61-s − 2·71-s + 2·79-s + 8·80-s + 9·81-s − 22·89-s + 28·95-s + 24·99-s − 30·101-s + 30·109-s − 34·121-s + 22·125-s + 127-s + 131-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s − 1.20·11-s + 16-s + 3.21·19-s + 25-s + 3.34·29-s − 5.02·31-s − 0.624·41-s − 1.78·45-s − 4/7·49-s − 1.07·55-s + 2.34·59-s + 1.79·61-s − 0.237·71-s + 0.225·79-s + 0.894·80-s + 81-s − 2.33·89-s + 2.87·95-s + 2.41·99-s − 2.98·101-s + 2.87·109-s − 3.09·121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(81450625s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(81450625s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 8145062581450625    =    54⋅1945^{4} \cdot 19^{4}
Sign: 11
Analytic conductor: 0.3311330.331133
Root analytic conductor: 0.8709640.870964
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 81450625, ( :1/2,1/2,1/2,1/2), 1)(8,\ 81450625,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.93258336300.9325833630
L(12)L(\frac12) ≈\approx 0.93258336300.9325833630
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)
bad5C22C_2^2 1−2T−T2−2pT3+p2T4 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4}
19C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
good2C22C_2^2×\timesC22C_2^2 (1−pT+pT2−p2T3+p2T4)(1+pT+pT2+p2T3+p2T4) ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )
3C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
7C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
11C2C_2 (1+T+pT2)4 ( 1 + T + p T^{2} )^{4}
13C22C_2^2×\timesC22C_2^2 (1−T2+p2T4)(1+23T2+p2T4) ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} )
17C22C_2^2×\timesC22C_2^2 (1−8T+47T2−8pT3+p2T4)(1+8T+47T2+8pT3+p2T4) ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )
23C23C_2^3 1+10T2−429T4+10p2T6+p4T8 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8}
29C22C_2^2 (1−9T+52T2−9pT3+p2T4)2 ( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}
31C2C_2 (1+7T+pT2)4 ( 1 + 7 T + p T^{2} )^{4}
37C2C_2 (1−12T+pT2)2(1+12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}
41C22C_2^2 (1+2T−37T2+2pT3+p2T4)2 ( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
43C23C_2^3 1+82T2+4875T4+82p2T6+p4T8 1 + 82 T^{2} + 4875 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8}
47C23C_2^3 1+58T2+1155T4+58p2T6+p4T8 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8}
53C22C_2^2×\timesC22C_2^2 (1−14T+143T2−14pT3+p2T4)(1+14T+143T2+14pT3+p2T4) ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} )
59C22C_2^2 (1−9T+22T2−9pT3+p2T4)2 ( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}
61C22C_2^2 (1−7T−12T2−7pT3+p2T4)2 ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2}
67C23C_2^3 1+34T2−3333T4+34p2T6+p4T8 1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8}
71C22C_2^2 (1+T−70T2+pT3+p2T4)2 ( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} )^{2}
73C22C_2^2×\timesC22C_2^2 (1−97T2+p2T4)(1+143T2+p2T4) ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} )
79C22C_2^2 (1−T−78T2−pT3+p2T4)2 ( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2}
83C22C_2^2 (1−130T2+p2T4)2 ( 1 - 130 T^{2} + p^{2} T^{4} )^{2}
89C22C_2^2 (1+11T+32T2+11pT3+p2T4)2 ( 1 + 11 T + 32 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2}
97C23C_2^3 1+158T2+15555T4+158p2T6+p4T8 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + 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

−10.28406807025431810497662859055, −10.19169022711037439401446761241, −9.679195793232533285153980504497, −9.532039439246845229554027544587, −9.398209563299963851999341288282, −8.806519243418086288951730869509, −8.537086427225120139224986603788, −8.479769171669099165672302364040, −7.997997415445342114601443402482, −7.83893619798098288182257797829, −7.22009183458066532805278103219, −6.99686756585096186919380791256, −6.98308444319837001016377361472, −6.24988958210879402343157399272, −5.66588674590193468099192679184, −5.64913526436897972261373285785, −5.40320872327279133319137090296, −5.20363376167158838329305437717, −4.88992968633206983173756073625, −3.99573480617704466686026911245, −3.37630375747422206201856635784, −3.23549390921444222657453093382, −2.76886642020995138513698274223, −2.32251101184469232458412983922, −1.27418545886597354390154600307, 1.27418545886597354390154600307, 2.32251101184469232458412983922, 2.76886642020995138513698274223, 3.23549390921444222657453093382, 3.37630375747422206201856635784, 3.99573480617704466686026911245, 4.88992968633206983173756073625, 5.20363376167158838329305437717, 5.40320872327279133319137090296, 5.64913526436897972261373285785, 5.66588674590193468099192679184, 6.24988958210879402343157399272, 6.98308444319837001016377361472, 6.99686756585096186919380791256, 7.22009183458066532805278103219, 7.83893619798098288182257797829, 7.997997415445342114601443402482, 8.479769171669099165672302364040, 8.537086427225120139224986603788, 8.806519243418086288951730869509, 9.398209563299963851999341288282, 9.532039439246845229554027544587, 9.679195793232533285153980504497, 10.19169022711037439401446761241, 10.28406807025431810497662859055

Graph of the ZZ-function along the critical line