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Properties

Label 4-1134e2-1.1-c1e2-0-26
Degree 44
Conductor 12859561285956
Sign 11
Analytic cond. 81.993681.9936
Root an. cond. 3.009153.00915
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-s − 3·5-s − 7-s − 8-s − 3·10-s + 3·11-s − 2·13-s − 14-s − 16-s − 3·17-s + 7·19-s + 3·22-s + 9·23-s + 5·25-s − 2·26-s + 6·29-s − 8·31-s − 3·34-s + 3·35-s + 37-s + 7·38-s + 3·40-s + 6·41-s − 2·43-s + 9·46-s − 6·49-s + 5·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.34·5-s − 0.377·7-s − 0.353·8-s − 0.948·10-s + 0.904·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s + 1.60·19-s + 0.639·22-s + 1.87·23-s + 25-s − 0.392·26-s + 1.11·29-s − 1.43·31-s − 0.514·34-s + 0.507·35-s + 0.164·37-s + 1.13·38-s + 0.474·40-s + 0.937·41-s − 0.304·43-s + 1.32·46-s − 6/7·49-s + 0.707·50-s + ⋯

Functional equation

Λ(s)=(1285956s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1285956s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12859561285956    =    22⋅38⋅722^{2} \cdot 3^{8} \cdot 7^{2}
Sign: 11
Analytic conductor: 81.993681.9936
Root analytic conductor: 3.009153.00915
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1285956, ( :1/2,1/2), 1)(4,\ 1285956,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8855958551.885595855
L(12)L(\frac12) ≈\approx 1.8855958551.885595855
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)
bad2C2C_2 1−T+T2 1 - T + T^{2}
3 1 1
7C2C_2 1+T+pT2 1 + T + p T^{2}
good5C22C_2^2 1+3T+4T2+3pT3+p2T4 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−3T−2T2−3pT3+p2T4 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4}
13C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
17C22C_2^2 1+3T−8T2+3pT3+p2T4 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
19C2C_2 (1−8T+pT2)(1+T+pT2) ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} )
23C22C_2^2 1−9T+58T2−9pT3+p2T4 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4}
29C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
31C22C_2^2 1+8T+33T2+8pT3+p2T4 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4}
37C2C_2 (1−11T+pT2)(1+10T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
43C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C22C_2^2 1+3T−44T2+3pT3+p2T4 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
61C22C_2^2 1+2T−57T2+2pT3+p2T4 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−4T−51T2−4pT3+p2T4 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C22C_2^2 1+11T+48T2+11pT3+p2T4 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−16T+177T2−16pT3+p2T4 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4}
83C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
89C22C_2^2 1+3T−80T2+3pT3+p2T4 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4}
97C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{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

−9.763371079700271750033845065730, −9.698159435431574631186755687159, −9.022242203062570169394821173371, −8.994483356206098040072409811059, −8.402216170363493012973867422393, −7.73850861012365371308687606563, −7.62880991567855686257310721311, −7.03176463761525780201726242480, −6.63105713034330587579294960008, −6.49296630974517161359268483236, −5.59403279720130216399926511526, −5.18042310440136433883196151622, −4.88888309057666382372367342116, −4.25072262521787065918735842529, −3.98369057993054870699753205744, −3.23075419705373495479617535382, −3.18771199687414758820583624022, −2.43325876126771577361709349264, −1.36262377345782059353876066622, −0.58518293903836423228401425110, 0.58518293903836423228401425110, 1.36262377345782059353876066622, 2.43325876126771577361709349264, 3.18771199687414758820583624022, 3.23075419705373495479617535382, 3.98369057993054870699753205744, 4.25072262521787065918735842529, 4.88888309057666382372367342116, 5.18042310440136433883196151622, 5.59403279720130216399926511526, 6.49296630974517161359268483236, 6.63105713034330587579294960008, 7.03176463761525780201726242480, 7.62880991567855686257310721311, 7.73850861012365371308687606563, 8.402216170363493012973867422393, 8.994483356206098040072409811059, 9.022242203062570169394821173371, 9.698159435431574631186755687159, 9.763371079700271750033845065730

Graph of the ZZ-function along the critical line