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Properties

Label 4-3528e2-1.1-c0e2-0-0
Degree 44
Conductor 1244678412446784
Sign 11
Analytic cond. 3.100063.10006
Root an. cond. 1.326911.32691
Motivic weight 00
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-s − 2·5-s + 6-s + 8-s + 2·10-s − 2·13-s + 2·15-s − 16-s + 19-s − 2·23-s − 24-s + 25-s + 2·26-s + 27-s − 2·30-s − 38-s + 2·39-s − 2·40-s + 2·46-s + 48-s − 50-s − 54-s − 57-s − 2·59-s + 61-s + 64-s + ⋯
L(s)  = 1  − 2-s − 3-s − 2·5-s + 6-s + 8-s + 2·10-s − 2·13-s + 2·15-s − 16-s + 19-s − 2·23-s − 24-s + 25-s + 2·26-s + 27-s − 2·30-s − 38-s + 2·39-s − 2·40-s + 2·46-s + 48-s − 50-s − 54-s − 57-s − 2·59-s + 61-s + 64-s + ⋯

Functional equation

Λ(s)=(12446784s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(12446784s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1244678412446784    =    26⋅34⋅742^{6} \cdot 3^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 3.100063.10006
Root analytic conductor: 1.326911.32691
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 12446784, ( :0,0), 1)(4,\ 12446784,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.00080323829880.0008032382988
L(12)L(\frac12) ≈\approx 0.00080323829880.0008032382988
L(1)L(1) 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}
3C2C_2 1+T+T2 1 + T + T^{2}
7 1 1
good5C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
11C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
13C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
17C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
19C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
23C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
29C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
31C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
37C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
41C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
43C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
47C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
53C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
59C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
61C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
67C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
71C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
73C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
79C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
83C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
89C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
97C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + 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

−8.678823072014999364670049501212, −8.605852893270088321251759999521, −8.018444713588179514722949124677, −7.88107479113781708540118643740, −7.56205853281525838035978625803, −7.19759269325710739546937599720, −7.05192716179111697690513014297, −6.44611536824688367775281278041, −5.77551784126851747780892134716, −5.71654111896407470516932760850, −5.01650576265853764239350981387, −4.74157142594756227394563381850, −4.31062343527203027913334724230, −4.12842748159661787608799036502, −3.50890274371951980866263204186, −3.04963581831301415464755874808, −2.42539684432091705781786939488, −1.83147568710794522010079156713, −1.03978766618099246977404762834, −0.02515613220077678073874183418, 0.02515613220077678073874183418, 1.03978766618099246977404762834, 1.83147568710794522010079156713, 2.42539684432091705781786939488, 3.04963581831301415464755874808, 3.50890274371951980866263204186, 4.12842748159661787608799036502, 4.31062343527203027913334724230, 4.74157142594756227394563381850, 5.01650576265853764239350981387, 5.71654111896407470516932760850, 5.77551784126851747780892134716, 6.44611536824688367775281278041, 7.05192716179111697690513014297, 7.19759269325710739546937599720, 7.56205853281525838035978625803, 7.88107479113781708540118643740, 8.018444713588179514722949124677, 8.605852893270088321251759999521, 8.678823072014999364670049501212

Graph of the ZZ-function along the critical line