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Properties

Label 4-1134e2-1.1-c1e2-0-42
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 − 4·5-s + 7-s + 8-s + 4·10-s − 2·11-s + 12·13-s − 14-s − 16-s − 2·17-s + 4·19-s + 2·22-s + 23-s + 5·25-s − 12·26-s + 8·29-s + 9·31-s + 2·34-s − 4·35-s − 8·37-s − 4·38-s − 4·40-s + 6·41-s + 4·43-s − 46-s + 9·47-s − 6·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.78·5-s + 0.377·7-s + 0.353·8-s + 1.26·10-s − 0.603·11-s + 3.32·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.426·22-s + 0.208·23-s + 25-s − 2.35·26-s + 1.48·29-s + 1.61·31-s + 0.342·34-s − 0.676·35-s − 1.31·37-s − 0.648·38-s − 0.632·40-s + 0.937·41-s + 0.609·43-s − 0.147·46-s + 1.31·47-s − 6/7·49-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 1.3436390261.343639026
L(12)L(\frac12) ≈\approx 1.3436390261.343639026
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+4T+11T2+4pT3+p2T4 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+2T−7T2+2pT3+p2T4 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
17C22C_2^2 1+2T−13T2+2pT3+p2T4 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4}
19C22C_2^2 1−4T−3T2−4pT3+p2T4 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−T−22T2−pT3+p2T4 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4}
29C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
31C22C_2^2 1−9T+50T2−9pT3+p2T4 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+8T+27T2+8pT3+p2T4 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4}
41C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
43C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
47C22C_2^2 1−9T+34T2−9pT3+p2T4 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−12T+91T2−12pT3+p2T4 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+4T−43T2+4pT3+p2T4 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+6T−25T2+6pT3+p2T4 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−14T+129T2−14pT3+p2T4 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4}
71C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
73C2C_2 (1−10T+pT2)(1+17T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} )
79C22C_2^2 1−3T−70T2−3pT3+p2T4 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4}
83C2C_2 (1−14T+pT2)2 ( 1 - 14 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 (1−10T+pT2)2 ( 1 - 10 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

−10.25737184758959382265477222857, −9.420141844727443610376752719788, −8.916149983533698404039515493193, −8.820794616823318769592956929817, −8.202532682907544812943221800991, −8.181077312189295711908508927066, −7.77924231238583808540504852395, −7.30154233033532838532509207918, −6.82105643504306638255227336053, −6.15213765450410592060512003283, −6.09185949045987581664774918540, −5.27086118095282339132565153868, −4.71096349616497830434552604123, −4.32458217783164232559932871444, −3.69568954276267644779935056547, −3.56859320293782943933975143561, −2.92101296086047482613191160163, −2.02650303208289799089492690495, −0.902563278206519705593890782152, −0.876661825813929335953256138876, 0.876661825813929335953256138876, 0.902563278206519705593890782152, 2.02650303208289799089492690495, 2.92101296086047482613191160163, 3.56859320293782943933975143561, 3.69568954276267644779935056547, 4.32458217783164232559932871444, 4.71096349616497830434552604123, 5.27086118095282339132565153868, 6.09185949045987581664774918540, 6.15213765450410592060512003283, 6.82105643504306638255227336053, 7.30154233033532838532509207918, 7.77924231238583808540504852395, 8.181077312189295711908508927066, 8.202532682907544812943221800991, 8.820794616823318769592956929817, 8.916149983533698404039515493193, 9.420141844727443610376752719788, 10.25737184758959382265477222857

Graph of the ZZ-function along the critical line