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Properties

Label 2-5225-1.1-c1-0-153
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  − 0.814·2-s + 2.50·3-s − 1.33·4-s − 2.03·6-s + 4.83·7-s + 2.71·8-s + 3.25·9-s − 11-s − 3.34·12-s + 4.12·13-s − 3.94·14-s + 0.459·16-s − 2.02·17-s − 2.64·18-s + 19-s + 12.0·21-s + 0.814·22-s − 6.38·23-s + 6.79·24-s − 3.35·26-s + 0.626·27-s − 6.46·28-s + 7.42·29-s − 4.40·31-s − 5.80·32-s − 2.50·33-s + 1.65·34-s + ⋯
L(s)  = 1  − 0.575·2-s + 1.44·3-s − 0.668·4-s − 0.831·6-s + 1.82·7-s + 0.960·8-s + 1.08·9-s − 0.301·11-s − 0.964·12-s + 1.14·13-s − 1.05·14-s + 0.114·16-s − 0.491·17-s − 0.624·18-s + 0.229·19-s + 2.64·21-s + 0.173·22-s − 1.33·23-s + 1.38·24-s − 0.658·26-s + 0.120·27-s − 1.22·28-s + 1.37·29-s − 0.791·31-s − 1.02·32-s − 0.435·33-s + 0.283·34-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 2.8293550432.829355043
L(12)L(\frac12) ≈\approx 2.8293550432.829355043
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+0.814T+2T2 1 + 0.814T + 2T^{2}
3 1−2.50T+3T2 1 - 2.50T + 3T^{2}
7 1−4.83T+7T2 1 - 4.83T + 7T^{2}
13 1−4.12T+13T2 1 - 4.12T + 13T^{2}
17 1+2.02T+17T2 1 + 2.02T + 17T^{2}
23 1+6.38T+23T2 1 + 6.38T + 23T^{2}
29 1−7.42T+29T2 1 - 7.42T + 29T^{2}
31 1+4.40T+31T2 1 + 4.40T + 31T^{2}
37 1+1.77T+37T2 1 + 1.77T + 37T^{2}
41 1−9.92T+41T2 1 - 9.92T + 41T^{2}
43 1−9.89T+43T2 1 - 9.89T + 43T^{2}
47 1+0.145T+47T2 1 + 0.145T + 47T^{2}
53 1−12.9T+53T2 1 - 12.9T + 53T^{2}
59 1−1.17T+59T2 1 - 1.17T + 59T^{2}
61 1−2.03T+61T2 1 - 2.03T + 61T^{2}
67 1+7.84T+67T2 1 + 7.84T + 67T^{2}
71 1+11.1T+71T2 1 + 11.1T + 71T^{2}
73 1−3.27T+73T2 1 - 3.27T + 73T^{2}
79 1−11.3T+79T2 1 - 11.3T + 79T^{2}
83 1+2.37T+83T2 1 + 2.37T + 83T^{2}
89 1+9.26T+89T2 1 + 9.26T + 89T^{2}
97 1+8.29T+97T2 1 + 8.29T + 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

−8.389559620063854657158146603372, −7.73397697735631548680163721061, −7.39941526449405982883314854952, −5.97650476619527079419425321649, −5.14988507173245207051569318239, −4.16830971656777077253310052524, −3.98210941745065109065857284253, −2.61785980887038576160409116741, −1.84200598797527007065576960172, −1.01813688135207007702235205849, 1.01813688135207007702235205849, 1.84200598797527007065576960172, 2.61785980887038576160409116741, 3.98210941745065109065857284253, 4.16830971656777077253310052524, 5.14988507173245207051569318239, 5.97650476619527079419425321649, 7.39941526449405982883314854952, 7.73397697735631548680163721061, 8.389559620063854657158146603372

Graph of the ZZ-function along the critical line