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Properties

Label 2-5225-1.1-c1-0-84
Degree 22
Conductor 52255225
Sign 11
Analytic cond. 41.721841.7218
Root an. cond. 6.459246.45924
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.37·2-s + 1.23·3-s − 0.115·4-s + 1.69·6-s − 2.43·7-s − 2.90·8-s − 1.47·9-s − 11-s − 0.142·12-s + 4.84·13-s − 3.33·14-s − 3.75·16-s + 1.04·17-s − 2.01·18-s + 19-s − 3.00·21-s − 1.37·22-s − 0.377·23-s − 3.59·24-s + 6.65·26-s − 5.52·27-s + 0.280·28-s + 4.50·29-s + 4.76·31-s + 0.652·32-s − 1.23·33-s + 1.42·34-s + ⋯
L(s)  = 1  + 0.970·2-s + 0.713·3-s − 0.0577·4-s + 0.692·6-s − 0.919·7-s − 1.02·8-s − 0.490·9-s − 0.301·11-s − 0.0412·12-s + 1.34·13-s − 0.892·14-s − 0.938·16-s + 0.252·17-s − 0.476·18-s + 0.229·19-s − 0.656·21-s − 0.292·22-s − 0.0786·23-s − 0.732·24-s + 1.30·26-s − 1.06·27-s + 0.0530·28-s + 0.836·29-s + 0.855·31-s + 0.115·32-s − 0.215·33-s + 0.244·34-s + ⋯

Functional equation

Λ(s)=(5225s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5225s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 52255225    =    52⋅11⋅195^{2} \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 41.721841.7218
Root analytic conductor: 6.459246.45924
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5225, ( :1/2), 1)(2,\ 5225,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9144616672.914461667
L(12)L(\frac12) ≈\approx 2.9144616672.914461667
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}
ppFp(T)F_p(T)
bad5 1 1
11 1+T 1 + T
19 1−T 1 - T
good2 1−1.37T+2T2 1 - 1.37T + 2T^{2}
3 1−1.23T+3T2 1 - 1.23T + 3T^{2}
7 1+2.43T+7T2 1 + 2.43T + 7T^{2}
13 1−4.84T+13T2 1 - 4.84T + 13T^{2}
17 1−1.04T+17T2 1 - 1.04T + 17T^{2}
23 1+0.377T+23T2 1 + 0.377T + 23T^{2}
29 1−4.50T+29T2 1 - 4.50T + 29T^{2}
31 1−4.76T+31T2 1 - 4.76T + 31T^{2}
37 1+3.01T+37T2 1 + 3.01T + 37T^{2}
41 1−0.790T+41T2 1 - 0.790T + 41T^{2}
43 1−7.46T+43T2 1 - 7.46T + 43T^{2}
47 1−9.67T+47T2 1 - 9.67T + 47T^{2}
53 1−12.7T+53T2 1 - 12.7T + 53T^{2}
59 1−1.32T+59T2 1 - 1.32T + 59T^{2}
61 1+5.58T+61T2 1 + 5.58T + 61T^{2}
67 1−1.20T+67T2 1 - 1.20T + 67T^{2}
71 1+0.259T+71T2 1 + 0.259T + 71T^{2}
73 1−6.27T+73T2 1 - 6.27T + 73T^{2}
79 1+9.16T+79T2 1 + 9.16T + 79T^{2}
83 1−14.2T+83T2 1 - 14.2T + 83T^{2}
89 1+1.23T+89T2 1 + 1.23T + 89T^{2}
97 1+2.95T+97T2 1 + 2.95T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.400474448760444489680562618472, −7.47771111024756857991928763615, −6.47875614586230447299819915545, −5.96159146123688397159616512443, −5.36232833314700783684007596902, −4.31135697734167401105240837139, −3.64153213588919939963380827169, −3.07974878828912326409243293296, −2.39941133230605221911362720594, −0.75514681611091505281583979759, 0.75514681611091505281583979759, 2.39941133230605221911362720594, 3.07974878828912326409243293296, 3.64153213588919939963380827169, 4.31135697734167401105240837139, 5.36232833314700783684007596902, 5.96159146123688397159616512443, 6.47875614586230447299819915545, 7.47771111024756857991928763615, 8.400474448760444489680562618472

Graph of the ZZ-function along the critical line