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Properties

Label 4-950e2-1.1-c1e2-0-9
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  + 2·2-s + 3-s + 3·4-s + 2·6-s + 7-s + 4·8-s − 9-s + 8·11-s + 3·12-s + 13-s + 2·14-s + 5·16-s − 11·17-s − 2·18-s − 2·19-s + 21-s + 16·22-s − 3·23-s + 4·24-s + 2·26-s + 3·28-s + 29-s − 2·31-s + 6·32-s + 8·33-s − 22·34-s − 3·36-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.816·6-s + 0.377·7-s + 1.41·8-s − 1/3·9-s + 2.41·11-s + 0.866·12-s + 0.277·13-s + 0.534·14-s + 5/4·16-s − 2.66·17-s − 0.471·18-s − 0.458·19-s + 0.218·21-s + 3.41·22-s − 0.625·23-s + 0.816·24-s + 0.392·26-s + 0.566·28-s + 0.185·29-s − 0.359·31-s + 1.06·32-s + 1.39·33-s − 3.77·34-s − 1/2·36-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 7.5263274027.526327402
L(12)L(\frac12) ≈\approx 7.5263274027.526327402
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)
bad2C1C_1 (1−T)2 ( 1 - T )^{2}
5 1 1
19C1C_1 (1+T)2 ( 1 + T )^{2}
good3D4D_{4} 1−T+2T2−pT3+p2T4 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4}
7D4D_{4} 1−T+10T2−pT3+p2T4 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4}
11C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
13C22C_2^2 1−T−12T2−pT3+p2T4 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4}
17D4D_{4} 1+11T+60T2+11pT3+p2T4 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+3T+10T2+3pT3+p2T4 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−T+20T2−pT3+p2T4 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4}
31D4D_{4} 1+2T+46T2+2pT3+p2T4 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4}
37C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
41D4D_{4} 1−8T+30T2−8pT3+p2T4 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−14T+118T2−14pT3+p2T4 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−4T+30T2−4pT3+p2T4 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−5T+108T2−5pT3+p2T4 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−T+114T2−pT3+p2T4 1 - T + 114 T^{2} - p T^{3} + p^{2} T^{4}
61D4D_{4} 1−14T+154T2−14pT3+p2T4 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−T+130T2−pT3+p2T4 1 - T + 130 T^{2} - p T^{3} + p^{2} T^{4}
71D4D_{4} 1−4T+78T2−4pT3+p2T4 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+9T+128T2+9pT3+p2T4 1 + 9 T + 128 T^{2} + 9 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+2T+142T2+2pT3+p2T4 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+14T+198T2+14pT3+p2T4 1 + 14 T + 198 T^{2} + 14 p T^{3} + p^{2} T^{4}
89C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
97C2C_2 (1+6T+pT2)2 ( 1 + 6 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

−10.40509723935053206255780021760, −9.730747983450564972363666958319, −9.269461802051136599333512278135, −9.030349635085689639443277468277, −8.607425023989674458271902987602, −8.218481723170876935839658217475, −7.63377684795684572540054953690, −7.00515204683514271804791357069, −6.80217316125886040642651172131, −6.25170554024009670678048785667, −6.06325101073125175662248675889, −5.56507900355847824074066298697, −4.67093250850662199388310576228, −4.34525009015419484453738364839, −3.94670328982411401578264404479, −3.90135760216719732222356982045, −2.76636362403408315057351541959, −2.48187998105717011530260448552, −1.88554258137734670923623349210, −1.06121728191310280520996406203, 1.06121728191310280520996406203, 1.88554258137734670923623349210, 2.48187998105717011530260448552, 2.76636362403408315057351541959, 3.90135760216719732222356982045, 3.94670328982411401578264404479, 4.34525009015419484453738364839, 4.67093250850662199388310576228, 5.56507900355847824074066298697, 6.06325101073125175662248675889, 6.25170554024009670678048785667, 6.80217316125886040642651172131, 7.00515204683514271804791357069, 7.63377684795684572540054953690, 8.218481723170876935839658217475, 8.607425023989674458271902987602, 9.030349635085689639443277468277, 9.269461802051136599333512278135, 9.730747983450564972363666958319, 10.40509723935053206255780021760

Graph of the ZZ-function along the critical line