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Properties

Label 4-1568e2-1.1-c1e2-0-16
Degree 44
Conductor 24586242458624
Sign 11
Analytic cond. 156.763156.763
Root an. cond. 3.538433.53843
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  + 2·5-s + 3·9-s + 12·13-s − 2·17-s + 5·25-s − 20·29-s + 2·37-s + 20·41-s + 6·45-s − 14·53-s + 10·61-s + 24·65-s + 6·73-s − 4·85-s − 10·89-s + 36·97-s + 2·101-s − 6·109-s − 28·113-s + 36·117-s + 11·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s − 40·145-s + ⋯
L(s)  = 1  + 0.894·5-s + 9-s + 3.32·13-s − 0.485·17-s + 25-s − 3.71·29-s + 0.328·37-s + 3.12·41-s + 0.894·45-s − 1.92·53-s + 1.28·61-s + 2.97·65-s + 0.702·73-s − 0.433·85-s − 1.05·89-s + 3.65·97-s + 0.199·101-s − 0.574·109-s − 2.63·113-s + 3.32·117-s + 121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.32·145-s + ⋯

Functional equation

Λ(s)=(2458624s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2458624s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 24586242458624    =    210⋅742^{10} \cdot 7^{4}
Sign: 11
Analytic conductor: 156.763156.763
Root analytic conductor: 3.538433.53843
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2458624, ( :1/2,1/2), 1)(4,\ 2458624,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.9286776103.928677610
L(12)L(\frac12) ≈\approx 3.9286776103.928677610
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
7 1 1
good3C2C_2 (1−pT+pT2)(1+pT+pT2) ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )
5C22C_2^2 1−2T−T2−2pT3+p2T4 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−pT2+p2T4 1 - p T^{2} + 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−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
23C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
29C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
31C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
37C22C_2^2 1−2T−33T2−2pT3+p2T4 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4}
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C22C_2^2 1+14T+143T2+14pT3+p2T4 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
61C22C_2^2 1−10T+39T2−10pT3+p2T4 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1−6T−37T2−6pT3+p2T4 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C22C_2^2 1+10T+11T2+10pT3+p2T4 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4}
97C2C_2 (1−18T+pT2)2 ( 1 - 18 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

−9.596217089251235973144226146753, −9.270842932231006894609393718444, −8.816278662756817364832643584961, −8.690156043388332829496223141848, −7.912591218713821630998378719785, −7.74772030784588976463769790506, −7.20508973447611266448702088031, −6.75283449306363278982894449113, −6.22360093555634451850504371113, −6.00123654017432615011133482722, −5.73116554112964815885542667805, −5.24281242106899192112193828474, −4.47188546696729274756658672390, −4.12494590271738282204569281943, −3.61633248902851439760982642349, −3.42891139092492171791587163282, −2.47314716694803096831033800656, −1.86345545908322150849026141964, −1.45286753045269302624972575186, −0.854463060309764483263012365521, 0.854463060309764483263012365521, 1.45286753045269302624972575186, 1.86345545908322150849026141964, 2.47314716694803096831033800656, 3.42891139092492171791587163282, 3.61633248902851439760982642349, 4.12494590271738282204569281943, 4.47188546696729274756658672390, 5.24281242106899192112193828474, 5.73116554112964815885542667805, 6.00123654017432615011133482722, 6.22360093555634451850504371113, 6.75283449306363278982894449113, 7.20508973447611266448702088031, 7.74772030784588976463769790506, 7.912591218713821630998378719785, 8.690156043388332829496223141848, 8.816278662756817364832643584961, 9.270842932231006894609393718444, 9.596217089251235973144226146753

Graph of the ZZ-function along the critical line