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Properties

Label 4-950e2-1.1-c3e2-0-1
Degree 44
Conductor 902500902500
Sign 11
Analytic cond. 3141.803141.80
Root an. cond. 7.486777.48677
Motivic weight 33
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  − 4·4-s + 38·9-s − 88·11-s + 16·16-s − 38·19-s + 324·29-s − 624·31-s − 152·36-s + 68·41-s + 352·44-s + 286·49-s − 728·59-s + 1.03e3·61-s − 64·64-s + 640·71-s + 152·76-s + 2.41e3·79-s + 715·81-s + 2.04e3·89-s − 3.34e3·99-s − 1.79e3·101-s − 2.09e3·109-s − 1.29e3·116-s + 3.14e3·121-s + 2.49e3·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.40·9-s − 2.41·11-s + 1/4·16-s − 0.458·19-s + 2.07·29-s − 3.61·31-s − 0.703·36-s + 0.259·41-s + 1.20·44-s + 0.833·49-s − 1.60·59-s + 2.17·61-s − 1/8·64-s + 1.06·71-s + 0.229·76-s + 3.44·79-s + 0.980·81-s + 2.43·89-s − 3.39·99-s − 1.76·101-s − 1.83·109-s − 1.03·116-s + 2.36·121-s + 1.80·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

Λ(s)=(902500s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(902500s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{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: 3141.803141.80
Root analytic conductor: 7.486777.48677
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 902500, ( :3/2,3/2), 1)(4,\ 902500,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.3122177001.312217700
L(12)L(\frac12) ≈\approx 1.3122177001.312217700
L(52)L(\frac{5}{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+p2T2 1 + p^{2} T^{2}
5 1 1
19C1C_1 (1+pT)2 ( 1 + p T )^{2}
good3C22C_2^2 1−38T2+p6T4 1 - 38 T^{2} + p^{6} T^{4}
7C22C_2^2 1−286T2+p6T4 1 - 286 T^{2} + p^{6} T^{4}
11C2C_2 (1+4pT+p3T2)2 ( 1 + 4 p T + p^{3} T^{2} )^{2}
13C22C_2^2 1−2630T2+p6T4 1 - 2630 T^{2} + p^{6} T^{4}
17C22C_2^2 1−2430T2+p6T4 1 - 2430 T^{2} + p^{6} T^{4}
23C22C_2^2 1+2562T2+p6T4 1 + 2562 T^{2} + p^{6} T^{4}
29C2C_2 (1−162T+p3T2)2 ( 1 - 162 T + p^{3} T^{2} )^{2}
31C2C_2 (1+312T+p3T2)2 ( 1 + 312 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−50230T2+p6T4 1 - 50230 T^{2} + p^{6} T^{4}
41C2C_2 (1−34T+p3T2)2 ( 1 - 34 T + p^{3} T^{2} )^{2}
43C22C_2^2 1+27610T2+p6T4 1 + 27610 T^{2} + p^{6} T^{4}
47C22C_2^2 1+128754T2+p6T4 1 + 128754 T^{2} + p^{6} T^{4}
53C22C_2^2 1−41718T2+p6T4 1 - 41718 T^{2} + p^{6} T^{4}
59C2C_2 (1+364T+p3T2)2 ( 1 + 364 T + p^{3} T^{2} )^{2}
61C2C_2 (1−518T+p3T2)2 ( 1 - 518 T + p^{3} T^{2} )^{2}
67C22C_2^2 1+252250T2+p6T4 1 + 252250 T^{2} + p^{6} T^{4}
71C2C_2 (1−320T+p3T2)2 ( 1 - 320 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−484270T2+p6T4 1 - 484270 T^{2} + p^{6} T^{4}
79C2C_2 (1−1208T+p3T2)2 ( 1 - 1208 T + p^{3} T^{2} )^{2}
83C22C_2^2 1+110826T2+p6T4 1 + 110826 T^{2} + p^{6} T^{4}
89C2C_2 (1−1022T+p3T2)2 ( 1 - 1022 T + p^{3} T^{2} )^{2}
97C22C_2^2 1−465790T2+p6T4 1 - 465790 T^{2} + p^{6} 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

−9.882721140005250697826044375749, −9.401091254797735589634204044887, −9.183769785699725115909318318797, −8.581470023089136208719360511137, −8.062665340537514203129447348305, −7.75009936945931162017649807694, −7.55368864538176865495321368123, −6.78130466521080935894303728682, −6.72177131456271208666227611312, −5.82886638171804293756282590521, −5.42394012778622198073007603905, −4.91598918117742613558666823765, −4.84685854497597558147563005285, −3.92651949796314032549333748919, −3.75363955329574058965639566199, −2.96897344424761143363456060342, −2.29956341570017540144513040906, −1.95274250460059837051992414129, −1.01310994925174928615743872480, −0.33415302523609919438880253279, 0.33415302523609919438880253279, 1.01310994925174928615743872480, 1.95274250460059837051992414129, 2.29956341570017540144513040906, 2.96897344424761143363456060342, 3.75363955329574058965639566199, 3.92651949796314032549333748919, 4.84685854497597558147563005285, 4.91598918117742613558666823765, 5.42394012778622198073007603905, 5.82886638171804293756282590521, 6.72177131456271208666227611312, 6.78130466521080935894303728682, 7.55368864538176865495321368123, 7.75009936945931162017649807694, 8.062665340537514203129447348305, 8.581470023089136208719360511137, 9.183769785699725115909318318797, 9.401091254797735589634204044887, 9.882721140005250697826044375749

Graph of the ZZ-function along the critical line