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Properties

Label 2-475-1.1-c1-0-1
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  − 0.816·2-s − 1.53·3-s − 1.33·4-s + 1.25·6-s − 5.03·7-s + 2.72·8-s − 0.633·9-s − 3.03·11-s + 2.05·12-s + 4.57·13-s + 4.11·14-s + 0.443·16-s + 1.07·17-s + 0.517·18-s + 19-s + 7.74·21-s + 2.47·22-s − 4.11·23-s − 4.18·24-s − 3.73·26-s + 5.58·27-s + 6.71·28-s − 1.07·29-s + 5.58·31-s − 5.80·32-s + 4.66·33-s − 0.879·34-s + ⋯
L(s)  = 1  − 0.577·2-s − 0.888·3-s − 0.666·4-s + 0.512·6-s − 1.90·7-s + 0.962·8-s − 0.211·9-s − 0.914·11-s + 0.592·12-s + 1.26·13-s + 1.09·14-s + 0.110·16-s + 0.261·17-s + 0.121·18-s + 0.229·19-s + 1.68·21-s + 0.528·22-s − 0.857·23-s − 0.854·24-s − 0.732·26-s + 1.07·27-s + 1.26·28-s − 0.199·29-s + 1.00·31-s − 1.02·32-s + 0.812·33-s − 0.150·34-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 0.35672255610.3567225561
L(12)L(\frac12) ≈\approx 0.35672255610.3567225561
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+0.816T+2T2 1 + 0.816T + 2T^{2}
3 1+1.53T+3T2 1 + 1.53T + 3T^{2}
7 1+5.03T+7T2 1 + 5.03T + 7T^{2}
11 1+3.03T+11T2 1 + 3.03T + 11T^{2}
13 1−4.57T+13T2 1 - 4.57T + 13T^{2}
17 1−1.07T+17T2 1 - 1.07T + 17T^{2}
23 1+4.11T+23T2 1 + 4.11T + 23T^{2}
29 1+1.07T+29T2 1 + 1.07T + 29T^{2}
31 1−5.58T+31T2 1 - 5.58T + 31T^{2}
37 1+0.0947T+37T2 1 + 0.0947T + 37T^{2}
41 1−10.6T+41T2 1 - 10.6T + 41T^{2}
43 1+5.03T+43T2 1 + 5.03T + 43T^{2}
47 1−12.2T+47T2 1 - 12.2T + 47T^{2}
53 1−4.09T+53T2 1 - 4.09T + 53T^{2}
59 1+1.39T+59T2 1 + 1.39T + 59T^{2}
61 1+5.69T+61T2 1 + 5.69T + 61T^{2}
67 1+5.28T+67T2 1 + 5.28T + 67T^{2}
71 1+5.67T+71T2 1 + 5.67T + 71T^{2}
73 1+9.07T+73T2 1 + 9.07T + 73T^{2}
79 1+5.39T+79T2 1 + 5.39T + 79T^{2}
83 1+1.95T+83T2 1 + 1.95T + 83T^{2}
89 1+2.18T+89T2 1 + 2.18T + 89T^{2}
97 1−2.16T+97T2 1 - 2.16T + 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

−10.63206226457689822574229513159, −10.23071942198849525938986948920, −9.292832878707652797997066644687, −8.495178541244963706394724501283, −7.37293913524197533799493645969, −6.13384615958499342549676490383, −5.67595031944522013313221045089, −4.21642320315638735892081658755, −3.02795612468554015068081966573, −0.60852986743194229708444574820, 0.60852986743194229708444574820, 3.02795612468554015068081966573, 4.21642320315638735892081658755, 5.67595031944522013313221045089, 6.13384615958499342549676490383, 7.37293913524197533799493645969, 8.495178541244963706394724501283, 9.292832878707652797997066644687, 10.23071942198849525938986948920, 10.63206226457689822574229513159

Graph of the ZZ-function along the critical line