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Properties

Label 8-525e4-1.1-c1e4-0-2
Degree 88
Conductor 7596914062575969140625
Sign 11
Analytic cond. 308.848308.848
Root an. cond. 2.047472.04747
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  − 2·4-s + 6·9-s − 5·16-s − 12·36-s − 24·41-s − 2·49-s − 48·59-s + 20·64-s − 32·79-s + 27·81-s + 24·89-s + 24·101-s + 8·109-s + 20·121-s + 127-s + 131-s + 137-s + 139-s − 30·144-s + 149-s + 151-s + 157-s + 163-s + 48·164-s + 167-s − 52·169-s + 173-s + ⋯
L(s)  = 1  − 4-s + 2·9-s − 5/4·16-s − 2·36-s − 3.74·41-s − 2/7·49-s − 6.24·59-s + 5/2·64-s − 3.60·79-s + 3·81-s + 2.54·89-s + 2.38·101-s + 0.766·109-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 3.74·164-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + ⋯

Functional equation

Λ(s)=((34⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(2−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(2-s)\end{aligned}
Λ(s)=((34⋅58⋅74)s/2 ΓC(s+1/2)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+1/2)^{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: 308.848308.848
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅58⋅74, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.75022624730.7502262473
L(12)L(\frac12) ≈\approx 0.75022624730.7502262473
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)
bad3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
5 1 1
7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
good2C22C_2^2 (1+T2+p2T4)2 ( 1 + T^{2} + p^{2} T^{4} )^{2}
11C22C_2^2 (1−10T2+p2T4)2 ( 1 - 10 T^{2} + p^{2} T^{4} )^{2}
13C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
17C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
19C2C_2 (1−8T+pT2)2(1+8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}
23C22C_2^2 (1+34T2+p2T4)2 ( 1 + 34 T^{2} + p^{2} T^{4} )^{2}
29C22C_2^2 (1−10T2+p2T4)2 ( 1 - 10 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
37C2C_2 (1−12T+pT2)2(1+12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}
41C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
43C22C_2^2 (1−22T2+p2T4)2 ( 1 - 22 T^{2} + p^{2} T^{4} )^{2}
47C22C_2^2 (1+50T2+p2T4)2 ( 1 + 50 T^{2} + p^{2} T^{4} )^{2}
53C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
59C2C_2 (1+12T+pT2)4 ( 1 + 12 T + p T^{2} )^{4}
61C2C_2 (1−14T+pT2)2(1+14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2}
67C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
71C22C_2^2 (1−130T2+p2T4)2 ( 1 - 130 T^{2} + p^{2} T^{4} )^{2}
73C22C_2^2 (1+98T2+p2T4)2 ( 1 + 98 T^{2} + p^{2} T^{4} )^{2}
79C2C_2 (1+8T+pT2)4 ( 1 + 8 T + p T^{2} )^{4}
83C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
89C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
97C22C_2^2 (1+146T2+p2T4)2 ( 1 + 146 T^{2} + p^{2} T^{4} )^{2}
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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

−7.80691236815812847801061710954, −7.57241858477325853407614587762, −7.26680344576942928881004047252, −7.11259579238074505442502198256, −7.05375632988217755079917383587, −6.70521078429031870970319935908, −6.32116898665438022856370985347, −6.11191443730594698930135763594, −6.04741784685578589167522285886, −5.72412556715800302836313090176, −4.95151852031948174821800403142, −4.87733816347142457872542983804, −4.84012215124110855288502951239, −4.75777086974369717831420958247, −4.40061393218100143317930118227, −3.96294151665143071234302213924, −3.84294620227438735573679925753, −3.39952301725385244109906433575, −3.07240870077918639526649588377, −2.97719156167324051890750767877, −2.16764245529004211572156235362, −1.77725026979383506704419911943, −1.75446713377922185508226898441, −1.19715780883625252829434054165, −0.29422615950307160400937029934, 0.29422615950307160400937029934, 1.19715780883625252829434054165, 1.75446713377922185508226898441, 1.77725026979383506704419911943, 2.16764245529004211572156235362, 2.97719156167324051890750767877, 3.07240870077918639526649588377, 3.39952301725385244109906433575, 3.84294620227438735573679925753, 3.96294151665143071234302213924, 4.40061393218100143317930118227, 4.75777086974369717831420958247, 4.84012215124110855288502951239, 4.87733816347142457872542983804, 4.95151852031948174821800403142, 5.72412556715800302836313090176, 6.04741784685578589167522285886, 6.11191443730594698930135763594, 6.32116898665438022856370985347, 6.70521078429031870970319935908, 7.05375632988217755079917383587, 7.11259579238074505442502198256, 7.26680344576942928881004047252, 7.57241858477325853407614587762, 7.80691236815812847801061710954

Graph of the ZZ-function along the critical line