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Properties

Label 2-5225-1.1-c1-0-178
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  + 1.28·2-s + 3.30·3-s − 0.336·4-s + 4.25·6-s + 0.563·7-s − 3.01·8-s + 7.89·9-s + 11-s − 1.10·12-s − 1.69·13-s + 0.726·14-s − 3.21·16-s − 0.0949·17-s + 10.1·18-s + 19-s + 1.86·21-s + 1.28·22-s + 0.625·23-s − 9.94·24-s − 2.19·26-s + 16.1·27-s − 0.189·28-s + 6.14·29-s + 3.11·31-s + 1.88·32-s + 3.30·33-s − 0.122·34-s + ⋯
L(s)  = 1  + 0.912·2-s + 1.90·3-s − 0.168·4-s + 1.73·6-s + 0.212·7-s − 1.06·8-s + 2.63·9-s + 0.301·11-s − 0.320·12-s − 0.471·13-s + 0.194·14-s − 0.803·16-s − 0.0230·17-s + 2.40·18-s + 0.229·19-s + 0.405·21-s + 0.275·22-s + 0.130·23-s − 2.03·24-s − 0.429·26-s + 3.10·27-s − 0.0358·28-s + 1.14·29-s + 0.559·31-s + 0.332·32-s + 0.574·33-s − 0.0210·34-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 6.0836865576.083686557
L(12)L(\frac12) ≈\approx 6.0836865576.083686557
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 1−1.28T+2T2 1 - 1.28T + 2T^{2}
3 1−3.30T+3T2 1 - 3.30T + 3T^{2}
7 1−0.563T+7T2 1 - 0.563T + 7T^{2}
13 1+1.69T+13T2 1 + 1.69T + 13T^{2}
17 1+0.0949T+17T2 1 + 0.0949T + 17T^{2}
23 1−0.625T+23T2 1 - 0.625T + 23T^{2}
29 1−6.14T+29T2 1 - 6.14T + 29T^{2}
31 1−3.11T+31T2 1 - 3.11T + 31T^{2}
37 1−2.91T+37T2 1 - 2.91T + 37T^{2}
41 1−0.562T+41T2 1 - 0.562T + 41T^{2}
43 1−10.9T+43T2 1 - 10.9T + 43T^{2}
47 1−8.90T+47T2 1 - 8.90T + 47T^{2}
53 1+0.489T+53T2 1 + 0.489T + 53T^{2}
59 1+11.5T+59T2 1 + 11.5T + 59T^{2}
61 1−13.8T+61T2 1 - 13.8T + 61T^{2}
67 1+2.05T+67T2 1 + 2.05T + 67T^{2}
71 1−3.06T+71T2 1 - 3.06T + 71T^{2}
73 1+5.61T+73T2 1 + 5.61T + 73T^{2}
79 1+5.37T+79T2 1 + 5.37T + 79T^{2}
83 1−2.47T+83T2 1 - 2.47T + 83T^{2}
89 1+14.0T+89T2 1 + 14.0T + 89T^{2}
97 1−6.13T+97T2 1 - 6.13T + 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

−8.290457101204523303021635981478, −7.59268145291499546728694673186, −6.87887437772439615281291649797, −6.00564064986472587900201730906, −4.91373634125869791728958088735, −4.33402098843267876097102487100, −3.74914553426473043209380193990, −2.86244839527890530189664005055, −2.44431525676286986894375510750, −1.14744459568178841947093311921, 1.14744459568178841947093311921, 2.44431525676286986894375510750, 2.86244839527890530189664005055, 3.74914553426473043209380193990, 4.33402098843267876097102487100, 4.91373634125869791728958088735, 6.00564064986472587900201730906, 6.87887437772439615281291649797, 7.59268145291499546728694673186, 8.290457101204523303021635981478

Graph of the ZZ-function along the critical line