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Properties

Label 8-525e4-1.1-c1e4-0-7
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·9-s − 4·11-s + 4·16-s + 4·19-s + 8·29-s − 2·36-s − 32·41-s − 4·44-s − 2·49-s + 12·59-s − 8·61-s + 11·64-s − 20·71-s + 4·76-s − 16·79-s + 3·81-s − 4·89-s + 8·99-s − 44·101-s − 28·109-s + 8·116-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1/2·4-s − 2/3·9-s − 1.20·11-s + 16-s + 0.917·19-s + 1.48·29-s − 1/3·36-s − 4.99·41-s − 0.603·44-s − 2/7·49-s + 1.56·59-s − 1.02·61-s + 11/8·64-s − 2.37·71-s + 0.458·76-s − 1.80·79-s + 1/3·81-s − 0.423·89-s + 0.804·99-s − 4.37·101-s − 2.68·109-s + 0.742·116-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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 1.1730440241.173044024
L(12)L(\frac12) ≈\approx 1.1730440241.173044024
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+T2)2 ( 1 + T^{2} )^{2}
5 1 1
7C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good2C23C_2^3 1−T2−3T4−p2T6+p4T8 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8}
11D4D_{4} (1+2T+3T2+2pT3+p2T4)2 ( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
13D4×C2D_4\times C_2 1−24T2+302T4−24p2T6+p4T8 1 - 24 T^{2} + 302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8}
17D4×C2D_4\times C_2 1+24T2+542T4+24p2T6+p4T8 1 + 24 T^{2} + 542 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8}
19D4D_{4} (1−2T+34T2−2pT3+p2T4)2 ( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
23C22C_2^2 (1−21T2+p2T4)2 ( 1 - 21 T^{2} + p^{2} T^{4} )^{2}
29D4D_{4} (1−4T+17T2−4pT3+p2T4)2 ( 1 - 4 T + 17 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+42T2+p2T4)2 ( 1 + 42 T^{2} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1−106T2+5467T4−106p2T6+p4T8 1 - 106 T^{2} + 5467 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8}
41C2C_2 (1+8T+pT2)4 ( 1 + 8 T + p T^{2} )^{4}
43D4×C2D_4\times C_2 1−90T2+5003T4−90p2T6+p4T8 1 - 90 T^{2} + 5003 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−128T2+8014T4−128p2T6+p4T8 1 - 128 T^{2} + 8014 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−140T2+9238T4−140p2T6+p4T8 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1−6T+122T2−6pT3+p2T4)2 ( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
61C4C_4 (1+4T−54T2+4pT3+p2T4)2 ( 1 + 4 T - 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−146T2+11427T4−146p2T6+p4T8 1 - 146 T^{2} + 11427 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1+10T+147T2+10pT3+p2T4)2 ( 1 + 10 T + 147 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1−280T2+30238T4−280p2T6+p4T8 1 - 280 T^{2} + 30238 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1+8T+169T2+8pT3+p2T4)2 ( 1 + 8 T + 169 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1−164T2+15382T4−164p2T6+p4T8 1 - 164 T^{2} + 15382 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8}
89D4D_{4} (1+2T+134T2+2pT3+p2T4)2 ( 1 + 2 T + 134 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−220T2+25798T4−220p2T6+p4T8 1 - 220 T^{2} + 25798 T^{4} - 220 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

−7.82779121250595269589007569860, −7.70529078176513594536598361853, −7.33474140706265203583111947204, −7.02135852238868284906466194689, −7.01504206603295341141920793164, −6.53698472330534938503928165471, −6.38209003437364867952162510224, −6.37372105154529693582361468244, −5.76125464253528751313768579938, −5.51662501056955730919185238440, −5.30313492292577676170906596520, −5.25222771584494571758207954451, −4.96767850459350227327135990783, −4.75178273847731043647911131995, −4.08066378282328285310519425593, −3.91637536216849347108536082304, −3.83966297596316986770140245596, −2.97834451304383027415082022637, −2.96822422112917751123288128406, −2.95626352174807229827986193188, −2.66502739460181134897732335368, −1.79005906929978794111508077973, −1.68028554892747885178977757722, −1.27679661244757547946785259237, −0.34176444194724140528034704113, 0.34176444194724140528034704113, 1.27679661244757547946785259237, 1.68028554892747885178977757722, 1.79005906929978794111508077973, 2.66502739460181134897732335368, 2.95626352174807229827986193188, 2.96822422112917751123288128406, 2.97834451304383027415082022637, 3.83966297596316986770140245596, 3.91637536216849347108536082304, 4.08066378282328285310519425593, 4.75178273847731043647911131995, 4.96767850459350227327135990783, 5.25222771584494571758207954451, 5.30313492292577676170906596520, 5.51662501056955730919185238440, 5.76125464253528751313768579938, 6.37372105154529693582361468244, 6.38209003437364867952162510224, 6.53698472330534938503928165471, 7.01504206603295341141920793164, 7.02135852238868284906466194689, 7.33474140706265203583111947204, 7.70529078176513594536598361853, 7.82779121250595269589007569860

Graph of the ZZ-function along the critical line