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Properties

Label 4-855e2-1.1-c1e2-0-14
Degree 44
Conductor 731025731025
Sign 11
Analytic cond. 46.610746.6107
Root an. cond. 2.612892.61289
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  + 3·4-s + 4·5-s + 4·11-s + 5·16-s − 2·19-s + 12·20-s + 11·25-s + 12·29-s − 8·31-s + 4·41-s + 12·44-s + 10·49-s + 16·55-s − 20·61-s + 3·64-s − 16·71-s − 6·76-s + 16·79-s + 20·80-s − 20·89-s − 8·95-s + 33·100-s + 12·109-s + 36·116-s − 10·121-s − 24·124-s + 24·125-s + ⋯
L(s)  = 1  + 3/2·4-s + 1.78·5-s + 1.20·11-s + 5/4·16-s − 0.458·19-s + 2.68·20-s + 11/5·25-s + 2.22·29-s − 1.43·31-s + 0.624·41-s + 1.80·44-s + 10/7·49-s + 2.15·55-s − 2.56·61-s + 3/8·64-s − 1.89·71-s − 0.688·76-s + 1.80·79-s + 2.23·80-s − 2.11·89-s − 0.820·95-s + 3.29·100-s + 1.14·109-s + 3.34·116-s − 0.909·121-s − 2.15·124-s + 2.14·125-s + ⋯

Functional equation

Λ(s)=(731025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(731025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 731025731025    =    34⋅52⋅1923^{4} \cdot 5^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 46.610746.6107
Root analytic conductor: 2.612892.61289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 731025, ( :1/2,1/2), 1)(4,\ 731025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.0090347435.009034743
L(12)L(\frac12) ≈\approx 5.0090347435.009034743
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)
bad3 1 1
5C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
19C1C_1 (1+T)2 ( 1 + T )^{2}
good2C22C_2^2 1−3T2+p2T4 1 - 3 T^{2} + p^{2} T^{4}
7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 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 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
43C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
89C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
97C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 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

−10.41737247471285469807786476730, −10.12646617100499277098909120431, −9.483567440917630241018606292112, −9.250840009713004995148678350012, −8.754137495052402914892306546719, −8.479230701203551187422644813196, −7.56024957916006174241903721531, −7.38306704759315663432768293714, −6.72892563003714813428130132180, −6.51362987178189988334855362282, −5.99831679603070817721854291951, −5.96564902743805933963082980578, −5.20257039028099728325110258959, −4.67836198328781943963638737601, −4.03065092351358301673726484637, −3.30432615157517333132330895532, −2.56970910005043211489291031175, −2.44436305421132028083560688713, −1.50865096095262405934657522825, −1.31120513869035294460168071989, 1.31120513869035294460168071989, 1.50865096095262405934657522825, 2.44436305421132028083560688713, 2.56970910005043211489291031175, 3.30432615157517333132330895532, 4.03065092351358301673726484637, 4.67836198328781943963638737601, 5.20257039028099728325110258959, 5.96564902743805933963082980578, 5.99831679603070817721854291951, 6.51362987178189988334855362282, 6.72892563003714813428130132180, 7.38306704759315663432768293714, 7.56024957916006174241903721531, 8.479230701203551187422644813196, 8.754137495052402914892306546719, 9.250840009713004995148678350012, 9.483567440917630241018606292112, 10.12646617100499277098909120431, 10.41737247471285469807786476730

Graph of the ZZ-function along the critical line