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Properties

Label 2-475-19.18-c2-0-50
Degree $2$
Conductor $475$
Sign $-0.884 + 0.466i$
Analytic cond. $12.9428$
Root an. cond. $3.59761$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.40i·2-s + 2.91i·3-s + 2.02·4-s + 4.10·6-s − 11.7·7-s − 8.46i·8-s + 0.493·9-s − 6.36·11-s + 5.89i·12-s − 17.7i·13-s + 16.4i·14-s − 3.82·16-s − 17.5·17-s − 0.694i·18-s + (−16.8 + 8.86i)19-s + ⋯
L(s)  = 1  − 0.703i·2-s + 0.972i·3-s + 0.505·4-s + 0.683·6-s − 1.67·7-s − 1.05i·8-s + 0.0548·9-s − 0.578·11-s + 0.491i·12-s − 1.36i·13-s + 1.17i·14-s − 0.238·16-s − 1.03·17-s − 0.0385i·18-s + (−0.884 + 0.466i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.884 + 0.466i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ -0.884 + 0.466i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6442680015\)
\(L(\frac12)\) \(\approx\) \(0.6442680015\)
\(L(2)\) 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 \( 1 + (16.8 - 8.86i)T \)
good2 \( 1 + 1.40iT - 4T^{2} \)
3 \( 1 - 2.91iT - 9T^{2} \)
7 \( 1 + 11.7T + 49T^{2} \)
11 \( 1 + 6.36T + 121T^{2} \)
13 \( 1 + 17.7iT - 169T^{2} \)
17 \( 1 + 17.5T + 289T^{2} \)
23 \( 1 - 1.53T + 529T^{2} \)
29 \( 1 + 39.9iT - 841T^{2} \)
31 \( 1 - 9.24iT - 961T^{2} \)
37 \( 1 + 56.4iT - 1.36e3T^{2} \)
41 \( 1 + 39.5iT - 1.68e3T^{2} \)
43 \( 1 - 18.4T + 1.84e3T^{2} \)
47 \( 1 + 61.5T + 2.20e3T^{2} \)
53 \( 1 - 66.5iT - 2.80e3T^{2} \)
59 \( 1 + 9.42iT - 3.48e3T^{2} \)
61 \( 1 + 36.5T + 3.72e3T^{2} \)
67 \( 1 + 2.55iT - 4.48e3T^{2} \)
71 \( 1 + 53.8iT - 5.04e3T^{2} \)
73 \( 1 + 115.T + 5.32e3T^{2} \)
79 \( 1 - 95.9iT - 6.24e3T^{2} \)
83 \( 1 - 143.T + 6.88e3T^{2} \)
89 \( 1 + 19.4iT - 7.92e3T^{2} \)
97 \( 1 - 163. iT - 9.40e3T^{2} \)
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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

−10.51346969906323295099932055841, −9.828822588659501735596092104401, −9.079884150721217275382922934221, −7.65300627395635361215276338919, −6.57713636407814042708557961567, −5.72263353069662408800171408293, −4.19667439442405872259766364178, −3.36550349997504367198763039328, −2.42961589380757118059857585439, −0.23352092089236466636240272445, 1.88004744362572440795243327576, 2.97133422237190877548330236880, 4.61977660985830581963689803062, 6.17283736383677509736860651195, 6.69791490585748433048838463144, 7.06535049982406274322796282465, 8.271225778378066320190056252473, 9.203990646942672900472176513373, 10.25893060671331993925174163085, 11.29948297376772314619064434494

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