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Properties

Label 4-245e2-1.1-c1e2-0-6
Degree 44
Conductor 6002560025
Sign 11
Analytic cond. 3.827243.82724
Root an. cond. 1.398691.39869
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  + 4·5-s + 5·9-s − 6·11-s − 4·16-s + 11·25-s + 10·29-s − 4·31-s − 4·41-s + 20·45-s − 24·55-s + 20·59-s + 16·61-s − 16·71-s + 10·79-s − 16·80-s + 16·81-s − 30·99-s − 24·101-s − 10·109-s + 5·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 20·144-s + 40·145-s + ⋯
L(s)  = 1  + 1.78·5-s + 5/3·9-s − 1.80·11-s − 16-s + 11/5·25-s + 1.85·29-s − 0.718·31-s − 0.624·41-s + 2.98·45-s − 3.23·55-s + 2.60·59-s + 2.04·61-s − 1.89·71-s + 1.12·79-s − 1.78·80-s + 16/9·81-s − 3.01·99-s − 2.38·101-s − 0.957·109-s + 5/11·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/3·144-s + 3.32·145-s + ⋯

Functional equation

Λ(s)=(60025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(60025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6002560025    =    52⋅745^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 3.827243.82724
Root analytic conductor: 1.398691.39869
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 60025, ( :1/2,1/2), 1)(4,\ 60025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9680308751.968030875
L(12)L(\frac12) ≈\approx 1.9680308751.968030875
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)
bad5C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
7 1 1
good2C2C_2 (1−pT+pT2)(1+pT+pT2) ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )
3C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
11C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
13C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
17C22C_2^2 1+15T2+p2T4 1 + 15 T^{2} + p^{2} T^{4}
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
31C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C22C_2^2 1−85T2+p2T4 1 - 85 T^{2} + p^{2} T^{4}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
83C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C22C_2^2 1−145T2+p2T4 1 - 145 T^{2} + p^{2} T^{4}
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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

−12.87800358646590012494933048348, −11.84499212526274276054351731312, −11.35436837004902089596449340197, −10.48830894694029595254842704192, −10.28435667535972885534450575044, −10.21239243694715159637043713959, −9.552653034685264689496698174365, −9.083276566823832879511463172761, −8.497055539128846024946525050910, −7.919503696406930494753125122485, −7.25555219626301205548661667912, −6.61974670488315296105460364924, −6.57030394030377988870854436187, −5.38402049899582686739742874953, −5.31323077445712023937989250955, −4.64975090428334336306619623857, −3.88899238689639897784968487792, −2.62782325971648773735830990284, −2.34216171330090280540722742525, −1.33621710577323121618905087609, 1.33621710577323121618905087609, 2.34216171330090280540722742525, 2.62782325971648773735830990284, 3.88899238689639897784968487792, 4.64975090428334336306619623857, 5.31323077445712023937989250955, 5.38402049899582686739742874953, 6.57030394030377988870854436187, 6.61974670488315296105460364924, 7.25555219626301205548661667912, 7.919503696406930494753125122485, 8.497055539128846024946525050910, 9.083276566823832879511463172761, 9.552653034685264689496698174365, 10.21239243694715159637043713959, 10.28435667535972885534450575044, 10.48830894694029595254842704192, 11.35436837004902089596449340197, 11.84499212526274276054351731312, 12.87800358646590012494933048348

Graph of the ZZ-function along the critical line