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Properties

Label 4-950e2-1.1-c1e2-0-3
Degree 44
Conductor 902500902500
Sign 11
Analytic cond. 57.544157.5441
Root an. cond. 2.754232.75423
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-s + 5·9-s − 12·11-s + 16-s − 2·19-s − 18·29-s − 8·31-s − 5·36-s + 12·44-s + 13·49-s − 18·59-s − 20·61-s − 64-s − 12·71-s + 2·76-s + 20·79-s + 16·81-s + 24·89-s − 60·99-s + 36·101-s − 22·109-s + 18·116-s + 86·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s + 5/3·9-s − 3.61·11-s + 1/4·16-s − 0.458·19-s − 3.34·29-s − 1.43·31-s − 5/6·36-s + 1.80·44-s + 13/7·49-s − 2.34·59-s − 2.56·61-s − 1/8·64-s − 1.42·71-s + 0.229·76-s + 2.25·79-s + 16/9·81-s + 2.54·89-s − 6.03·99-s + 3.58·101-s − 2.10·109-s + 1.67·116-s + 7.81·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

Λ(s)=(902500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(902500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 902500902500    =    22⋅54⋅1922^{2} \cdot 5^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 57.544157.5441
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 902500, ( :1/2,1/2), 1)(4,\ 902500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.57852540960.5785254096
L(12)L(\frac12) ≈\approx 0.57852540960.5785254096
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+T2 1 + T^{2}
5 1 1
19C1C_1 (1+T)2 ( 1 + T )^{2}
good3C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
11C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
13C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
17C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
23C22C_2^2 1−37T2+p2T4 1 - 37 T^{2} + p^{2} T^{4}
29C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
31C2C_2 (1+4T+pT2)2 ( 1 + 4 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+pT2)2 ( 1 + p T^{2} )^{2}
43C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C22C_2^2 1−97T2+p2T4 1 - 97 T^{2} + p^{2} T^{4}
59C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C22C_2^2 1−109T2+p2T4 1 - 109 T^{2} + p^{2} T^{4}
71C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
73C22C_2^2 1−97T2+p2T4 1 - 97 T^{2} + p^{2} T^{4}
79C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
83C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
97C22C_2^2 1−94T2+p2T4 1 - 94 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

−10.42424864242956759141114593136, −9.876017528067997263728010117759, −9.363353566030147453465344299148, −9.139184014635326537383955715586, −8.660162121413821916064949680429, −7.77612921283094186400315734487, −7.72984663965283997999488472376, −7.36968635601476475027561036474, −7.36655446060614346946881406783, −6.18205371221264052512287134121, −5.94588866048212482367291290365, −5.20581549300400466359991490832, −5.15246677623423518985594486824, −4.60657500121310258618406628634, −4.01447318220950203180659533846, −3.48119750726470909916394202472, −2.89482614234160538226307188203, −2.04124522009189983557643337644, −1.83976765163521167928633654641, −0.33900283762510085792816425473, 0.33900283762510085792816425473, 1.83976765163521167928633654641, 2.04124522009189983557643337644, 2.89482614234160538226307188203, 3.48119750726470909916394202472, 4.01447318220950203180659533846, 4.60657500121310258618406628634, 5.15246677623423518985594486824, 5.20581549300400466359991490832, 5.94588866048212482367291290365, 6.18205371221264052512287134121, 7.36655446060614346946881406783, 7.36968635601476475027561036474, 7.72984663965283997999488472376, 7.77612921283094186400315734487, 8.660162121413821916064949680429, 9.139184014635326537383955715586, 9.363353566030147453465344299148, 9.876017528067997263728010117759, 10.42424864242956759141114593136

Graph of the ZZ-function along the critical line