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Properties

Label 2-475-1.1-c1-0-4
Degree 22
Conductor 475475
Sign 11
Analytic cond. 3.792893.79289
Root an. cond. 1.947531.94753
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  − 0.246·2-s − 0.801·3-s − 1.93·4-s + 0.198·6-s + 1.69·7-s + 0.972·8-s − 2.35·9-s − 0.911·11-s + 1.55·12-s + 1.55·13-s − 0.417·14-s + 3.63·16-s + 5.29·17-s + 0.582·18-s − 19-s − 1.35·21-s + 0.225·22-s + 4.24·23-s − 0.780·24-s − 0.384·26-s + 4.29·27-s − 3.28·28-s + 5.00·29-s + 1.82·31-s − 2.84·32-s + 0.731·33-s − 1.30·34-s + ⋯
L(s)  = 1  − 0.174·2-s − 0.462·3-s − 0.969·4-s + 0.0808·6-s + 0.639·7-s + 0.343·8-s − 0.785·9-s − 0.274·11-s + 0.448·12-s + 0.431·13-s − 0.111·14-s + 0.909·16-s + 1.28·17-s + 0.137·18-s − 0.229·19-s − 0.296·21-s + 0.0480·22-s + 0.885·23-s − 0.159·24-s − 0.0753·26-s + 0.826·27-s − 0.620·28-s + 0.930·29-s + 0.328·31-s − 0.502·32-s + 0.127·33-s − 0.224·34-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 11
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 475, ( :1/2), 1)(2,\ 475,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.90541117660.9054111766
L(12)L(\frac12) ≈\approx 0.90541117660.9054111766
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
19 1+T 1 + T
good2 1+0.246T+2T2 1 + 0.246T + 2T^{2}
3 1+0.801T+3T2 1 + 0.801T + 3T^{2}
7 1−1.69T+7T2 1 - 1.69T + 7T^{2}
11 1+0.911T+11T2 1 + 0.911T + 11T^{2}
13 1−1.55T+13T2 1 - 1.55T + 13T^{2}
17 1−5.29T+17T2 1 - 5.29T + 17T^{2}
23 1−4.24T+23T2 1 - 4.24T + 23T^{2}
29 1−5.00T+29T2 1 - 5.00T + 29T^{2}
31 1−1.82T+31T2 1 - 1.82T + 31T^{2}
37 1−6.29T+37T2 1 - 6.29T + 37T^{2}
41 1−4.18T+41T2 1 - 4.18T + 41T^{2}
43 1−7.31T+43T2 1 - 7.31T + 43T^{2}
47 1+2.04T+47T2 1 + 2.04T + 47T^{2}
53 1+2.70T+53T2 1 + 2.70T + 53T^{2}
59 1−9.87T+59T2 1 - 9.87T + 59T^{2}
61 1−0.542T+61T2 1 - 0.542T + 61T^{2}
67 1+13.9T+67T2 1 + 13.9T + 67T^{2}
71 1+12.8T+71T2 1 + 12.8T + 71T^{2}
73 1+2.80T+73T2 1 + 2.80T + 73T^{2}
79 1−1.59T+79T2 1 - 1.59T + 79T^{2}
83 1−12.2T+83T2 1 - 12.2T + 83T^{2}
89 1−2.91T+89T2 1 - 2.91T + 89T^{2}
97 1−1.55T+97T2 1 - 1.55T + 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

−10.94964268241966714381145074329, −10.17807471108975932271771237286, −9.136005947361075249441790902075, −8.345102264079253792791639000254, −7.62669815188220381188037319731, −6.11045748435607278069387818145, −5.29469188982789780528686237166, −4.42912433186574816172859778924, −3.03624469790924194472196530940, −0.973117189691318859222470061420, 0.973117189691318859222470061420, 3.03624469790924194472196530940, 4.42912433186574816172859778924, 5.29469188982789780528686237166, 6.11045748435607278069387818145, 7.62669815188220381188037319731, 8.345102264079253792791639000254, 9.136005947361075249441790902075, 10.17807471108975932271771237286, 10.94964268241966714381145074329

Graph of the ZZ-function along the critical line