Properties

Label 2-475-1.1-c1-0-16
Degree $2$
Conductor $475$
Sign $1$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·2-s − 0.537·3-s + 3.85·4-s − 1.29·6-s + 3.18·7-s + 4.49·8-s − 2.71·9-s + 4.15·11-s − 2.07·12-s − 2.07·13-s + 7.71·14-s + 3.15·16-s − 5.79·17-s − 6.56·18-s − 19-s − 1.71·21-s + 10.0·22-s + 2.60·23-s − 2.41·24-s − 5.01·26-s + 3.06·27-s + 12.2·28-s + 6·29-s + 2.59·31-s − 1.34·32-s − 2.23·33-s − 14.0·34-s + ⋯
L(s)  = 1  + 1.71·2-s − 0.310·3-s + 1.92·4-s − 0.530·6-s + 1.20·7-s + 1.58·8-s − 0.903·9-s + 1.25·11-s − 0.597·12-s − 0.574·13-s + 2.06·14-s + 0.788·16-s − 1.40·17-s − 1.54·18-s − 0.229·19-s − 0.373·21-s + 2.14·22-s + 0.543·23-s − 0.492·24-s − 0.982·26-s + 0.590·27-s + 2.32·28-s + 1.11·29-s + 0.466·31-s − 0.237·32-s − 0.388·33-s − 2.40·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.456026870\)
\(L(\frac12)\) \(\approx\) \(3.456026870\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + T \)
good2 \( 1 - 2.41T + 2T^{2} \)
3 \( 1 + 0.537T + 3T^{2} \)
7 \( 1 - 3.18T + 7T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
13 \( 1 + 2.07T + 13T^{2} \)
17 \( 1 + 5.79T + 17T^{2} \)
23 \( 1 - 2.60T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 2.59T + 31T^{2} \)
37 \( 1 + 4.30T + 37T^{2} \)
41 \( 1 + 0.599T + 41T^{2} \)
43 \( 1 + 3.18T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + 1.71T + 59T^{2} \)
61 \( 1 + 8.75T + 61T^{2} \)
67 \( 1 + 4.76T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 - 2.72T + 73T^{2} \)
79 \( 1 + 1.40T + 79T^{2} \)
83 \( 1 - 7.07T + 83T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 + 2.07T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.42914549877807920452444824913, −10.72963131554115289684461111801, −9.125612398491009934797537683860, −8.161361446199647684270679948434, −6.77589657797894483872649228642, −6.23747713927914624437019652328, −4.91908694619561831623350060779, −4.61746627410580203861480523527, −3.23585636369928182955420610606, −1.93948977006853465589612621317, 1.93948977006853465589612621317, 3.23585636369928182955420610606, 4.61746627410580203861480523527, 4.91908694619561831623350060779, 6.23747713927914624437019652328, 6.77589657797894483872649228642, 8.161361446199647684270679948434, 9.125612398491009934797537683860, 10.72963131554115289684461111801, 11.42914549877807920452444824913

Graph of the $Z$-function along the critical line