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Properties

Label 4-416e2-1.1-c1e2-0-1
Degree 44
Conductor 173056173056
Sign 11
Analytic cond. 11.034211.0342
Root an. cond. 1.822571.82257
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  − 6·9-s − 6·13-s − 4·17-s − 6·25-s + 20·29-s + 14·49-s + 28·53-s − 20·61-s + 27·81-s − 4·101-s + 28·113-s + 36·117-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 23·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2·9-s − 1.66·13-s − 0.970·17-s − 6/5·25-s + 3.71·29-s + 2·49-s + 3.84·53-s − 2.56·61-s + 3·81-s − 0.398·101-s + 2.63·113-s + 3.32·117-s + 2·121-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 + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

Λ(s)=(173056s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(173056s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 173056173056    =    210⋅1322^{10} \cdot 13^{2}
Sign: 11
Analytic conductor: 11.034211.0342
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 173056, ( :1/2,1/2), 1)(4,\ 173056,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.95341673050.9534167305
L(12)L(\frac12) ≈\approx 0.95341673050.9534167305
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
13C2C_2 1+6T+pT2 1 + 6 T + p T^{2}
good3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
31C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
59C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
71C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
89C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} )
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.55787075739832438918349982840, −10.94623992581870519931806295802, −10.57524265039742480878316034732, −9.980054978256597878816658108354, −9.851700634749097054144291123639, −8.835093893370694411023005394471, −8.775739352324658909156762102108, −8.473765674485950046188673942610, −7.70294448963979100495083754388, −7.33458006870500264489307943622, −6.73761233265253247376963409325, −6.14192674121894524829240884087, −5.80091516039162686028993999232, −5.14439042862379674767591967660, −4.67785442856069665614237807109, −4.12631740983543554866834252820, −3.18000753413123720366118804601, −2.44773161809234098578861073100, −2.42921245897540854870854579688, −0.60630896463668980869045042343, 0.60630896463668980869045042343, 2.42921245897540854870854579688, 2.44773161809234098578861073100, 3.18000753413123720366118804601, 4.12631740983543554866834252820, 4.67785442856069665614237807109, 5.14439042862379674767591967660, 5.80091516039162686028993999232, 6.14192674121894524829240884087, 6.73761233265253247376963409325, 7.33458006870500264489307943622, 7.70294448963979100495083754388, 8.473765674485950046188673942610, 8.775739352324658909156762102108, 8.835093893370694411023005394471, 9.851700634749097054144291123639, 9.980054978256597878816658108354, 10.57524265039742480878316034732, 10.94623992581870519931806295802, 11.55787075739832438918349982840

Graph of the ZZ-function along the critical line