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Properties

Label 8-90e8-1.1-c1e4-0-2
Degree 88
Conductor 4.305×10154.305\times 10^{15}
Sign 11
Analytic cond. 1.75004×1071.75004\times 10^{7}
Root an. cond. 8.042318.04231
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  + 4·19-s − 24·29-s − 4·31-s + 24·41-s + 20·49-s − 24·59-s − 16·61-s + 24·71-s + 16·79-s + 24·101-s + 4·109-s − 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 0.917·19-s − 4.45·29-s − 0.718·31-s + 3.74·41-s + 20/7·49-s − 3.12·59-s − 2.04·61-s + 2.84·71-s + 1.80·79-s + 2.38·101-s + 0.383·109-s − 3.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

Λ(s)=((28⋅316⋅58)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((28⋅316⋅58)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28⋅316⋅582^{8} \cdot 3^{16} \cdot 5^{8}
Sign: 11
Analytic conductor: 1.75004×1071.75004\times 10^{7}
Root analytic conductor: 8.042318.04231
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅316⋅58, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{8} \cdot 3^{16} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.4576079461.457607946
L(12)L(\frac12) ≈\approx 1.4576079461.457607946
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)
bad2 1 1
3 1 1
5 1 1
good7D4×C2D_4\times C_2 1−20T2+186T4−20p2T6+p4T8 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
11C22C_2^2 (1+19T2+p2T4)2 ( 1 + 19 T^{2} + p^{2} T^{4} )^{2}
13D4×C2D_4\times C_2 1−20T2+246T4−20p2T6+p4T8 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
17D4×C2D_4\times C_2 1−44T2+954T4−44p2T6+p4T8 1 - 44 T^{2} + 954 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}
19D4D_{4} (1−2T+27T2−2pT3+p2T4)2 ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
23C22C_2^2 (1−34T2+p2T4)2 ( 1 - 34 T^{2} + p^{2} T^{4} )^{2}
29D4D_{4} (1+12T+91T2+12pT3+p2T4)2 ( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
31D4D_{4} (1+2T+15T2+2pT3+p2T4)2 ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1−92T2+4746T4−92p2T6+p4T8 1 - 92 T^{2} + 4746 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8}
41D4D_{4} (1−12T+91T2−12pT3+p2T4)2 ( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1−116T2+6762T4−116p2T6+p4T8 1 - 116 T^{2} + 6762 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−164T2+11034T4−164p2T6+p4T8 1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−44T2+5130T4−44p2T6+p4T8 1 - 44 T^{2} + 5130 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1+12T+151T2+12pT3+p2T4)2 ( 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
61C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
67D4×C2D_4\times C_2 1−20T2+2166T4−20p2T6+p4T8 1 - 20 T^{2} + 2166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1−12T+151T2−12pT3+p2T4)2 ( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1−188T2+16794T4−188p2T6+p4T8 1 - 188 T^{2} + 16794 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1−8T+66T2−8pT3+p2T4)2 ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1−20T2+8586T4−20p2T6+p4T8 1 - 20 T^{2} + 8586 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
89C22C_2^2 (1+151T2+p2T4)2 ( 1 + 151 T^{2} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−380T2+54906T4−380p2T6+p4T8 1 - 380 T^{2} + 54906 T^{4} - 380 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

−5.57102585267548162113678575558, −5.43373247788282536580030267928, −5.14229331087035426504742197099, −4.83742244922653844135233795420, −4.76016505369932647820947080055, −4.72223964476221415109475682988, −4.38056870104588391142390359373, −4.00897604053955738836722710170, −3.85160082984493484948385184900, −3.83965080749752124721595933566, −3.75579654970022471664609909864, −3.41975297551013825950267292812, −3.34179194150733243741301246630, −2.99951132303552983284904893186, −2.71610249868806172257141623291, −2.43565002958516511346958203896, −2.28987874419423450495269966416, −2.25194070438566794164048200024, −2.02333429146801692597178282020, −1.59060936046664083177389947721, −1.29255634186707623467288052427, −1.09441886584919454489348562437, −1.08561613217301888949976881122, −0.39250421923980820701298727275, −0.19290917447958255108560666359, 0.19290917447958255108560666359, 0.39250421923980820701298727275, 1.08561613217301888949976881122, 1.09441886584919454489348562437, 1.29255634186707623467288052427, 1.59060936046664083177389947721, 2.02333429146801692597178282020, 2.25194070438566794164048200024, 2.28987874419423450495269966416, 2.43565002958516511346958203896, 2.71610249868806172257141623291, 2.99951132303552983284904893186, 3.34179194150733243741301246630, 3.41975297551013825950267292812, 3.75579654970022471664609909864, 3.83965080749752124721595933566, 3.85160082984493484948385184900, 4.00897604053955738836722710170, 4.38056870104588391142390359373, 4.72223964476221415109475682988, 4.76016505369932647820947080055, 4.83742244922653844135233795420, 5.14229331087035426504742197099, 5.43373247788282536580030267928, 5.57102585267548162113678575558

Graph of the ZZ-function along the critical line