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Properties

Label 4-504e2-1.1-c1e2-0-45
Degree 44
Conductor 254016254016
Sign −1-1
Analytic cond. 16.196216.1962
Root an. cond. 2.006102.00610
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 2·13-s + 8·19-s − 2·25-s − 12·31-s + 12·43-s + 9·49-s − 10·61-s + 4·67-s − 2·73-s + 28·79-s + 8·91-s − 22·97-s − 28·103-s − 16·109-s − 16·121-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.554·13-s + 1.83·19-s − 2/5·25-s − 2.15·31-s + 1.82·43-s + 9/7·49-s − 1.28·61-s + 0.488·67-s − 0.234·73-s + 3.15·79-s + 0.838·91-s − 2.23·97-s − 2.75·103-s − 1.53·109-s − 1.45·121-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-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 − 1.69·169-s + ⋯

Functional equation

Λ(s)=(254016s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(254016s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 254016254016    =    26⋅34⋅722^{6} \cdot 3^{4} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 16.196216.1962
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 254016, ( :1/2,1/2), −1)(4,\ 254016,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
3 1 1
7C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
good5C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1+pT2)(1+2T+pT2) ( 1 + p T^{2} )( 1 + 2 T + p T^{2} )
17C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
29C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1+4T+pT2)(1+8T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−8T+pT2)(1−4T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} )
47C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
53C22C_2^2 1+96T2+p2T4 1 + 96 T^{2} + p^{2} T^{4}
59C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−12T+pT2)(1+8T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C22C_2^2 1−48T2+p2T4 1 - 48 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−14T+pT2)(1+16T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−16T+pT2)(1−12T+pT2) ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} )
83C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
89C22C_2^2 1−138T2+p2T4 1 - 138 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1+6T+pT2)(1+16T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 16 T + p 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

−9.043462667056503806108705241629, −8.003599324063389811057123014271, −7.79878616964519320582268691818, −7.19604479377758133271602972121, −6.90075289897899598594459069842, −6.33714351407957963113019189643, −5.67190630256624796310263274860, −5.46337069349078717428764603789, −4.82302141503652744836727592228, −3.82337619004569276644530500587, −3.71611000857255139890066142245, −2.89971891583939525270261787002, −2.42448426830572615930005105990, −1.27584716179387089883907185902, 0, 1.27584716179387089883907185902, 2.42448426830572615930005105990, 2.89971891583939525270261787002, 3.71611000857255139890066142245, 3.82337619004569276644530500587, 4.82302141503652744836727592228, 5.46337069349078717428764603789, 5.67190630256624796310263274860, 6.33714351407957963113019189643, 6.90075289897899598594459069842, 7.19604479377758133271602972121, 7.79878616964519320582268691818, 8.003599324063389811057123014271, 9.043462667056503806108705241629

Graph of the ZZ-function along the critical line