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Properties

Label 4-45e2-1.1-c1e2-0-1
Degree 44
Conductor 20252025
Sign 11
Analytic cond. 0.1291150.129115
Root an. cond. 0.5994380.599438
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 3·16-s − 8·19-s − 5·25-s + 16·31-s + 14·49-s + 4·61-s + 7·64-s + 8·76-s − 32·79-s + 5·100-s + 28·109-s − 22·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1/2·4-s − 3/4·16-s − 1.83·19-s − 25-s + 2.87·31-s + 2·49-s + 0.512·61-s + 7/8·64-s + 0.917·76-s − 3.60·79-s + 1/2·100-s + 2.68·109-s − 2·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

Λ(s)=(2025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20252025    =    34⋅523^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 0.1291150.129115
Root analytic conductor: 0.5994380.599438
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2025, ( :1/2,1/2), 1)(4,\ 2025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.58697321690.5869732169
L(12)L(\frac12) ≈\approx 0.58697321690.5869732169
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)
bad3 1 1
5C2C_2 1+pT2 1 + p T^{2}
good2C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
7C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
47C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
53C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
79C2C_2 (1+16T+pT2)2 ( 1 + 16 T + p T^{2} )^{2}
83C22C_2^2 1+154T2+p2T4 1 + 154 T^{2} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1−pT2)2 ( 1 - 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

−16.04425621665513637432335881101, −15.39860978254693558799803182116, −15.31086366215355683397844908499, −14.18964599311887806872946724336, −14.03829554572186610771548305855, −13.15723379959748243640876161274, −12.96574043565923018826779476430, −11.92843380880703408712681539674, −11.66876266699699984465493727952, −10.71802248997556069743590768513, −10.18166894925267983021659200493, −9.570850536528072839705758819793, −8.529513714199935388900375216623, −8.506199327349381512983870230050, −7.36447877954871795658687378464, −6.52566376180141557577636495203, −5.84873109781954041807523169102, −4.63076666346438307078059690242, −4.10424053282794606423576741067, −2.49439517148849264400889437608, 2.49439517148849264400889437608, 4.10424053282794606423576741067, 4.63076666346438307078059690242, 5.84873109781954041807523169102, 6.52566376180141557577636495203, 7.36447877954871795658687378464, 8.506199327349381512983870230050, 8.529513714199935388900375216623, 9.570850536528072839705758819793, 10.18166894925267983021659200493, 10.71802248997556069743590768513, 11.66876266699699984465493727952, 11.92843380880703408712681539674, 12.96574043565923018826779476430, 13.15723379959748243640876161274, 14.03829554572186610771548305855, 14.18964599311887806872946724336, 15.31086366215355683397844908499, 15.39860978254693558799803182116, 16.04425621665513637432335881101

Graph of the ZZ-function along the critical line