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Properties

Label 4-504e2-1.1-c1e2-0-14
Degree 44
Conductor 254016254016
Sign 11
Analytic cond. 16.196216.1962
Root an. cond. 2.006102.00610
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-s − 4-s + 3·8-s + 8·11-s − 16-s − 8·22-s − 10·25-s − 5·32-s + 24·43-s − 8·44-s − 7·49-s + 10·50-s + 7·64-s + 8·67-s − 24·86-s + 24·88-s + 7·98-s + 10·100-s + 40·107-s + 4·113-s + 26·121-s + 127-s + 3·128-s + 131-s − 8·134-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2.41·11-s − 1/4·16-s − 1.70·22-s − 2·25-s − 0.883·32-s + 3.65·43-s − 1.20·44-s − 49-s + 1.41·50-s + 7/8·64-s + 0.977·67-s − 2.58·86-s + 2.55·88-s + 0.707·98-s + 100-s + 3.86·107-s + 0.376·113-s + 2.36·121-s + 0.0887·127-s + 0.265·128-s + 0.0873·131-s − 0.691·134-s + 0.0854·137-s + 0.0848·139-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: 11
Analytic conductor: 16.196216.1962
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 254016, ( :1/2,1/2), 1)(4,\ 254016,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1654309101.165430910
L(12)L(\frac12) ≈\approx 1.1654309101.165430910
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)
bad2C2C_2 1+T+pT2 1 + T + p T^{2}
3 1 1
7C2C_2 1+pT2 1 + p T^{2}
good5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
19C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
43C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
61C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
73C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
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

−11.20961909287264512335255494150, −10.68269158535820971643676472078, −9.967431838943688016049105126496, −9.791990885502652367938957204653, −9.214816801595176800766717039881, −9.121566261915184788051845778858, −8.625903801365671790590457509253, −8.036103551643299844878937876337, −7.57992102978520621740334554130, −7.21622080974161626884662623880, −6.59623147254894945359251602149, −5.99081122971328745865502444770, −5.80431869535159251952096146703, −4.85560911127170109792540992087, −4.31315715256686274165587589135, −3.89518728377610978818893024479, −3.53264502831397871108854111783, −2.31471174308058104799078855800, −1.59107872405238015477568672891, −0.806097709578990692063738973092, 0.806097709578990692063738973092, 1.59107872405238015477568672891, 2.31471174308058104799078855800, 3.53264502831397871108854111783, 3.89518728377610978818893024479, 4.31315715256686274165587589135, 4.85560911127170109792540992087, 5.80431869535159251952096146703, 5.99081122971328745865502444770, 6.59623147254894945359251602149, 7.21622080974161626884662623880, 7.57992102978520621740334554130, 8.036103551643299844878937876337, 8.625903801365671790590457509253, 9.121566261915184788051845778858, 9.214816801595176800766717039881, 9.791990885502652367938957204653, 9.967431838943688016049105126496, 10.68269158535820971643676472078, 11.20961909287264512335255494150

Graph of the ZZ-function along the critical line