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Properties

Label 2-475-19.18-c2-0-11
Degree $2$
Conductor $475$
Sign $-0.836 + 0.547i$
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  + 1.47i·2-s + 3.55i·3-s + 1.82·4-s − 5.24·6-s − 7.05·7-s + 8.58i·8-s − 3.65·9-s + 2.24·11-s + 6.50i·12-s + 6.86i·13-s − 10.4i·14-s − 5.34·16-s − 11.1·17-s − 5.38i·18-s + (−15.8 + 10.4i)19-s + ⋯
L(s)  = 1  + 0.736i·2-s + 1.18i·3-s + 0.457·4-s − 0.873·6-s − 1.00·7-s + 1.07i·8-s − 0.406·9-s + 0.203·11-s + 0.542i·12-s + 0.527i·13-s − 0.743i·14-s − 0.333·16-s − 0.658·17-s − 0.299i·18-s + (−0.836 + 0.547i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.836 + 0.547i$
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: \((2,\ 475,\ (\ :1),\ -0.836 + 0.547i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.311854181\)
\(L(\frac12)\) \(\approx\) \(1.311854181\)
\(L(2)\) 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 \( 1 + (15.8 - 10.4i)T \)
good2 \( 1 - 1.47iT - 4T^{2} \)
3 \( 1 - 3.55iT - 9T^{2} \)
7 \( 1 + 7.05T + 49T^{2} \)
11 \( 1 - 2.24T + 121T^{2} \)
13 \( 1 - 6.86iT - 169T^{2} \)
17 \( 1 + 11.1T + 289T^{2} \)
23 \( 1 + 21.1T + 529T^{2} \)
29 \( 1 + 45.9iT - 841T^{2} \)
31 \( 1 - 14.7iT - 961T^{2} \)
37 \( 1 + 25.0iT - 1.36e3T^{2} \)
41 \( 1 - 64.9iT - 1.68e3T^{2} \)
43 \( 1 - 44.0T + 1.84e3T^{2} \)
47 \( 1 - 34.0T + 2.20e3T^{2} \)
53 \( 1 + 20.8iT - 2.80e3T^{2} \)
59 \( 1 - 69.2iT - 3.48e3T^{2} \)
61 \( 1 - 23.4T + 3.72e3T^{2} \)
67 \( 1 + 79.3iT - 4.48e3T^{2} \)
71 \( 1 - 26.8iT - 5.04e3T^{2} \)
73 \( 1 - 99.3T + 5.32e3T^{2} \)
79 \( 1 + 109. iT - 6.24e3T^{2} \)
83 \( 1 + 124.T + 6.88e3T^{2} \)
89 \( 1 - 85.7iT - 7.92e3T^{2} \)
97 \( 1 + 35.5iT - 9.40e3T^{2} \)
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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

−11.13838751462809542098245645076, −10.31272746582851090327148718295, −9.572964667004823729178424210461, −8.713684420407269065604431986256, −7.61802169486997077622078541602, −6.49082185011748052251028419121, −5.96068719067938272004881276403, −4.60700038087051305782348125533, −3.72277827033392470177323967147, −2.30690443968567608658763565727, 0.49283892928324950187905616457, 1.88412285438986453679976859910, 2.83688323490417857157973504263, 4.03887454516569796823110450999, 5.88365287717323976359747528520, 6.73355811247475592171508615744, 7.21260956355013353256722654407, 8.444041880743617132848605891630, 9.548247629813159464749365439680, 10.44945379707104132422363945427

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