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Properties

Label 2-5225-1.1-c1-0-120
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  − 2.22·2-s + 1.47·3-s + 2.96·4-s − 3.29·6-s + 4.42·7-s − 2.15·8-s − 0.812·9-s − 11-s + 4.38·12-s + 3.92·13-s − 9.86·14-s − 1.13·16-s + 1.61·17-s + 1.81·18-s + 19-s + 6.54·21-s + 2.22·22-s − 0.113·23-s − 3.18·24-s − 8.75·26-s − 5.63·27-s + 13.1·28-s + 1.08·29-s − 4.17·31-s + 6.83·32-s − 1.47·33-s − 3.59·34-s + ⋯
L(s)  = 1  − 1.57·2-s + 0.853·3-s + 1.48·4-s − 1.34·6-s + 1.67·7-s − 0.762·8-s − 0.270·9-s − 0.301·11-s + 1.26·12-s + 1.08·13-s − 2.63·14-s − 0.282·16-s + 0.391·17-s + 0.427·18-s + 0.229·19-s + 1.42·21-s + 0.475·22-s − 0.0236·23-s − 0.650·24-s − 1.71·26-s − 1.08·27-s + 2.48·28-s + 0.201·29-s − 0.749·31-s + 1.20·32-s − 0.257·33-s − 0.617·34-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 1.5632492481.563249248
L(12)L(\frac12) ≈\approx 1.5632492481.563249248
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+2.22T+2T2 1 + 2.22T + 2T^{2}
3 1−1.47T+3T2 1 - 1.47T + 3T^{2}
7 1−4.42T+7T2 1 - 4.42T + 7T^{2}
13 1−3.92T+13T2 1 - 3.92T + 13T^{2}
17 1−1.61T+17T2 1 - 1.61T + 17T^{2}
23 1+0.113T+23T2 1 + 0.113T + 23T^{2}
29 1−1.08T+29T2 1 - 1.08T + 29T^{2}
31 1+4.17T+31T2 1 + 4.17T + 31T^{2}
37 1−5.75T+37T2 1 - 5.75T + 37T^{2}
41 1−4.26T+41T2 1 - 4.26T + 41T^{2}
43 1+3.62T+43T2 1 + 3.62T + 43T^{2}
47 1−6.39T+47T2 1 - 6.39T + 47T^{2}
53 1+12.0T+53T2 1 + 12.0T + 53T^{2}
59 1−0.883T+59T2 1 - 0.883T + 59T^{2}
61 1−3.77T+61T2 1 - 3.77T + 61T^{2}
67 1−12.2T+67T2 1 - 12.2T + 67T^{2}
71 1−4.28T+71T2 1 - 4.28T + 71T^{2}
73 1−1.92T+73T2 1 - 1.92T + 73T^{2}
79 1+7.81T+79T2 1 + 7.81T + 79T^{2}
83 1−3.33T+83T2 1 - 3.33T + 83T^{2}
89 1+11.9T+89T2 1 + 11.9T + 89T^{2}
97 1−13.3T+97T2 1 - 13.3T + 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

−8.336590790983827096246798319118, −7.78327708317457051889673808428, −7.37155386237562960494025484193, −6.23611808900795718302292538494, −5.39706959678756601572935967398, −4.46476869028552449083531091576, −3.46542153791902390847598288143, −2.40782520568370742672461272781, −1.73295852480627462967350414203, −0.871098282183441681635004060858, 0.871098282183441681635004060858, 1.73295852480627462967350414203, 2.40782520568370742672461272781, 3.46542153791902390847598288143, 4.46476869028552449083531091576, 5.39706959678756601572935967398, 6.23611808900795718302292538494, 7.37155386237562960494025484193, 7.78327708317457051889673808428, 8.336590790983827096246798319118

Graph of the ZZ-function along the critical line