Properties

Label 2-475-95.94-c2-0-48
Degree $2$
Conductor $475$
Sign $0.834 + 0.550i$
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  + 2.36·2-s + 3.55·3-s + 1.59·4-s + 8.41·6-s − 11.0i·7-s − 5.68·8-s + 3.64·9-s + 16.7·11-s + 5.67·12-s + 14.3·13-s − 26.2i·14-s − 19.8·16-s + 20.2i·17-s + 8.61·18-s + (16.4 − 9.51i)19-s + ⋯
L(s)  = 1  + 1.18·2-s + 1.18·3-s + 0.399·4-s + 1.40·6-s − 1.58i·7-s − 0.710·8-s + 0.404·9-s + 1.52·11-s + 0.472·12-s + 1.10·13-s − 1.87i·14-s − 1.23·16-s + 1.18i·17-s + 0.478·18-s + (0.865 − 0.500i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.834 + 0.550i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (474, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ 0.834 + 0.550i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.509017629\)
\(L(\frac12)\) \(\approx\) \(4.509017629\)
\(L(2)\) 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 \( 1 + (-16.4 + 9.51i)T \)
good2 \( 1 - 2.36T + 4T^{2} \)
3 \( 1 - 3.55T + 9T^{2} \)
7 \( 1 + 11.0iT - 49T^{2} \)
11 \( 1 - 16.7T + 121T^{2} \)
13 \( 1 - 14.3T + 169T^{2} \)
17 \( 1 - 20.2iT - 289T^{2} \)
23 \( 1 + 13.5iT - 529T^{2} \)
29 \( 1 + 30.3iT - 841T^{2} \)
31 \( 1 - 16.1iT - 961T^{2} \)
37 \( 1 + 51.2T + 1.36e3T^{2} \)
41 \( 1 - 74.6iT - 1.68e3T^{2} \)
43 \( 1 - 6.23iT - 1.84e3T^{2} \)
47 \( 1 - 44.0iT - 2.20e3T^{2} \)
53 \( 1 + 30.8T + 2.80e3T^{2} \)
59 \( 1 - 73.0iT - 3.48e3T^{2} \)
61 \( 1 + 17.7T + 3.72e3T^{2} \)
67 \( 1 - 20.5T + 4.48e3T^{2} \)
71 \( 1 - 7.48iT - 5.04e3T^{2} \)
73 \( 1 + 29.2iT - 5.32e3T^{2} \)
79 \( 1 - 8.78iT - 6.24e3T^{2} \)
83 \( 1 + 63.7iT - 6.88e3T^{2} \)
89 \( 1 - 162. iT - 7.92e3T^{2} \)
97 \( 1 + 75.5T + 9.40e3T^{2} \)
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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

−10.87439608615519423374362452406, −9.699868156438467496968495404699, −8.868710781019655674690995820851, −8.037408638501114212492733389942, −6.83487055237883070410681639220, −6.08542378204333094811279812975, −4.44172604972249008167695389333, −3.83656391234237997816145768107, −3.17474129289160893625890809721, −1.33566989988918416899440497892, 1.94480024079726895637264055987, 3.21799921073433383809034457720, 3.69805031361308105518659788939, 5.19935872589614372390611492360, 5.91654993239465284403665544112, 7.03319991911698628228074406637, 8.523688270177540018409384917684, 8.999637742429494532375650229985, 9.523767475558605233877531863737, 11.41049475686453082814242490531

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