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Properties

Label 4-350e2-1.1-c1e2-0-16
Degree 44
Conductor 122500122500
Sign 11
Analytic cond. 7.810707.81070
Root an. cond. 1.671751.67175
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  − 2-s + 3-s − 6-s + 7-s + 8-s + 3·9-s + 6·11-s + 8·13-s − 14-s − 16-s − 3·18-s − 2·19-s + 21-s − 6·22-s − 3·23-s + 24-s − 8·26-s + 8·27-s − 6·29-s − 8·31-s + 6·33-s − 4·37-s + 2·38-s + 8·39-s + 18·41-s − 42-s + 14·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 9-s + 1.80·11-s + 2.21·13-s − 0.267·14-s − 1/4·16-s − 0.707·18-s − 0.458·19-s + 0.218·21-s − 1.27·22-s − 0.625·23-s + 0.204·24-s − 1.56·26-s + 1.53·27-s − 1.11·29-s − 1.43·31-s + 1.04·33-s − 0.657·37-s + 0.324·38-s + 1.28·39-s + 2.81·41-s − 0.154·42-s + 2.13·43-s + ⋯

Functional equation

Λ(s)=(122500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(122500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 122500122500    =    22⋅54⋅722^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 7.810707.81070
Root analytic conductor: 1.671751.67175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 122500, ( :1/2,1/2), 1)(4,\ 122500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7777860661.777786066
L(12)L(\frac12) ≈\approx 1.7777860661.777786066
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+T2 1 + T + T^{2}
5 1 1
7C2C_2 1−T+pT2 1 - T + p T^{2}
good3C22C_2^2 1−T−2T2−pT3+p2T4 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4}
11C22C_2^2 1−6T+25T2−6pT3+p2T4 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
17C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
19C22C_2^2 1+2T−15T2+2pT3+p2T4 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+3T−14T2+3pT3+p2T4 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4}
29C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
31C22C_2^2 1+8T+33T2+8pT3+p2T4 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+4T−21T2+4pT3+p2T4 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4}
41C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
43C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C22C_2^2 1+6T−17T2+6pT3+p2T4 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−6T−23T2−6pT3+p2T4 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+5T−36T2+5pT3+p2T4 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4}
67C2C_2 (1−16T+pT2)(1+11T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} )
71C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
73C22C_2^2 1+16T+183T2+16pT3+p2T4 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4}
79C22C_2^2 1+2T−75T2+2pT3+p2T4 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4}
83C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
89C22C_2^2 1−15T+136T2−15pT3+p2T4 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4}
97C2C_2 (1+14T+pT2)2 ( 1 + 14 T + 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

−11.44531500776178172422114943476, −11.14633118662792361884377904949, −10.80464514412187505352138054054, −10.40390860085647571050119551459, −9.646881200135278145608348813606, −9.189683285765849672773553132719, −9.003532390468188680012755658569, −8.708747181862142582449605468436, −7.918198123693549389279408758495, −7.73052133363762198184662013333, −6.99614545356602163056626935680, −6.56068623140851519642977026679, −5.99059187620313463522781212705, −5.53934080598421419226546320249, −4.36038292048665956218834927761, −3.95193641127438853822237163575, −3.87121871979297141974644355668, −2.71238625560086399033449734165, −1.46375846141625953745599028381, −1.37271047232170476058201585725, 1.37271047232170476058201585725, 1.46375846141625953745599028381, 2.71238625560086399033449734165, 3.87121871979297141974644355668, 3.95193641127438853822237163575, 4.36038292048665956218834927761, 5.53934080598421419226546320249, 5.99059187620313463522781212705, 6.56068623140851519642977026679, 6.99614545356602163056626935680, 7.73052133363762198184662013333, 7.918198123693549389279408758495, 8.708747181862142582449605468436, 9.003532390468188680012755658569, 9.189683285765849672773553132719, 9.646881200135278145608348813606, 10.40390860085647571050119551459, 10.80464514412187505352138054054, 11.14633118662792361884377904949, 11.44531500776178172422114943476

Graph of the ZZ-function along the critical line