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Properties

Label 8-867e4-1.1-c1e4-0-7
Degree 88
Conductor 565036352721565036352721
Sign 11
Analytic cond. 2297.122297.12
Root an. cond. 2.631162.63116
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  + 8·4-s + 4·13-s + 40·16-s + 24·47-s + 32·52-s + 160·64-s − 16·67-s − 81-s + 20·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + 181-s + 192·188-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 4·4-s + 1.10·13-s + 10·16-s + 3.50·47-s + 4.43·52-s + 20·64-s − 1.95·67-s − 1/9·81-s + 1.97·103-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 − 3.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 14.0·188-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

Λ(s)=((34⋅178)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((34⋅178)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅1783^{4} \cdot 17^{8}
Sign: 11
Analytic conductor: 2297.122297.12
Root analytic conductor: 2.631162.63116
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅178, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 17^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 15.0288677815.02886778
L(12)L(\frac12) ≈\approx 15.0288677815.02886778
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)
bad3C22C_2^2 1+T4 1 + T^{4}
17 1 1
good2C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
5C23C_2^3 1−49T4+p4T8 1 - 49 T^{4} + p^{4} T^{8}
7C23C_2^3 1−94T4+p4T8 1 - 94 T^{4} + p^{4} T^{8}
11C23C_2^3 1−73T4+p4T8 1 - 73 T^{4} + p^{4} T^{8}
13C2C_2 (1−T+pT2)4 ( 1 - T + p T^{2} )^{4}
19C22C_2^2 (1−37T2+p2T4)2 ( 1 - 37 T^{2} + p^{2} T^{4} )^{2}
23C23C_2^3 1+167T4+p4T8 1 + 167 T^{4} + p^{4} T^{8}
29C23C_2^3 1−1198T4+p4T8 1 - 1198 T^{4} + p^{4} T^{8}
31C23C_2^3 1+1442T4+p4T8 1 + 1442 T^{4} + p^{4} T^{8}
37C23C_2^3 1+626T4+p4T8 1 + 626 T^{4} + p^{4} T^{8}
41C23C_2^3 1+1967T4+p4T8 1 + 1967 T^{4} + p^{4} T^{8}
43C22C_2^2 (1−37T2+p2T4)2 ( 1 - 37 T^{2} + p^{2} T^{4} )^{2}
47C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
53C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1−82T2+p2T4)2 ( 1 - 82 T^{2} + p^{2} T^{4} )^{2}
61C23C_2^3 1−4078T4+p4T8 1 - 4078 T^{4} + p^{4} T^{8}
67C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
71C23C_2^3 1−10078T4+p4T8 1 - 10078 T^{4} + p^{4} T^{8}
73C22C_2^2×\timesC22C_2^2 (1−24T+288T2−24pT3+p2T4)(1+24T+288T2+24pT3+p2T4) ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} )
79C23C_2^3 1−9118T4+p4T8 1 - 9118 T^{4} + p^{4} T^{8}
83C22C_2^2 (1−130T2+p2T4)2 ( 1 - 130 T^{2} + p^{2} T^{4} )^{2}
89C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
97C23C_2^3 1−14974T4+p4T8 1 - 14974 T^{4} + 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

−7.21066038262213218237400892302, −7.05227474450183067054582144474, −7.00448533804454035030760724083, −6.57043846800013997310569913738, −6.41328134376657296504451112001, −6.02825478777099626855272140848, −6.02496317141060105256383324225, −5.92568082293885500060104968465, −5.67709935703679624950240077231, −5.43742112121643895585287664639, −5.09657817114600806386762546556, −4.63167706717331166262869664550, −4.52533707973565570620088048009, −3.87869288877347300655668065745, −3.74444128469671896603943371645, −3.48893680702854685641541075935, −3.39528092979712272246478327571, −2.83102171316810189339802850338, −2.67955178810694344772057441194, −2.43594688520711333077623419212, −2.28909075395878645021437729485, −1.70001946305155502000231631962, −1.65255687833407967701485660321, −1.06395671732091897442757008800, −0.945693231677351886500393808523, 0.945693231677351886500393808523, 1.06395671732091897442757008800, 1.65255687833407967701485660321, 1.70001946305155502000231631962, 2.28909075395878645021437729485, 2.43594688520711333077623419212, 2.67955178810694344772057441194, 2.83102171316810189339802850338, 3.39528092979712272246478327571, 3.48893680702854685641541075935, 3.74444128469671896603943371645, 3.87869288877347300655668065745, 4.52533707973565570620088048009, 4.63167706717331166262869664550, 5.09657817114600806386762546556, 5.43742112121643895585287664639, 5.67709935703679624950240077231, 5.92568082293885500060104968465, 6.02496317141060105256383324225, 6.02825478777099626855272140848, 6.41328134376657296504451112001, 6.57043846800013997310569913738, 7.00448533804454035030760724083, 7.05227474450183067054582144474, 7.21066038262213218237400892302

Graph of the ZZ-function along the critical line