Properties

Label 2-475-19.18-c2-0-36
Degree $2$
Conductor $475$
Sign $0.865 + 0.500i$
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  + 2.36i·2-s − 3.55i·3-s − 1.59·4-s + 8.41·6-s − 11.0·7-s + 5.68i·8-s − 3.64·9-s + 16.7·11-s + 5.67i·12-s − 14.3i·13-s − 26.2i·14-s − 19.8·16-s + 20.2·17-s − 8.61i·18-s + (−16.4 − 9.51i)19-s + ⋯
L(s)  = 1  + 1.18i·2-s − 1.18i·3-s − 0.399·4-s + 1.40·6-s − 1.58·7-s + 0.710i·8-s − 0.404·9-s + 1.52·11-s + 0.472i·12-s − 1.10i·13-s − 1.87i·14-s − 1.23·16-s + 1.18·17-s − 0.478i·18-s + (−0.865 − 0.500i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.865 + 0.500i$
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: \((2,\ 475,\ (\ :1),\ 0.865 + 0.500i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.526389086\)
\(L(\frac12)\) \(\approx\) \(1.526389086\)
\(L(2)\) 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 \( 1 - 2.36iT - 4T^{2} \)
3 \( 1 + 3.55iT - 9T^{2} \)
7 \( 1 + 11.0T + 49T^{2} \)
11 \( 1 - 16.7T + 121T^{2} \)
13 \( 1 + 14.3iT - 169T^{2} \)
17 \( 1 - 20.2T + 289T^{2} \)
23 \( 1 - 13.5T + 529T^{2} \)
29 \( 1 + 30.3iT - 841T^{2} \)
31 \( 1 + 16.1iT - 961T^{2} \)
37 \( 1 + 51.2iT - 1.36e3T^{2} \)
41 \( 1 + 74.6iT - 1.68e3T^{2} \)
43 \( 1 + 6.23T + 1.84e3T^{2} \)
47 \( 1 - 44.0T + 2.20e3T^{2} \)
53 \( 1 - 30.8iT - 2.80e3T^{2} \)
59 \( 1 - 73.0iT - 3.48e3T^{2} \)
61 \( 1 + 17.7T + 3.72e3T^{2} \)
67 \( 1 - 20.5iT - 4.48e3T^{2} \)
71 \( 1 + 7.48iT - 5.04e3T^{2} \)
73 \( 1 - 29.2T + 5.32e3T^{2} \)
79 \( 1 - 8.78iT - 6.24e3T^{2} \)
83 \( 1 - 63.7T + 6.88e3T^{2} \)
89 \( 1 - 162. iT - 7.92e3T^{2} \)
97 \( 1 + 75.5iT - 9.40e3T^{2} \)
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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

−10.68127940979448827587876920287, −9.519068140990672644488372981217, −8.684376474981555082976825121722, −7.55480703705317134643084771864, −7.03495884908785397307755283337, −6.22812646600217082474746739212, −5.73128298500213874278397949594, −3.87481146581601675584597158302, −2.47175550392047726328496129001, −0.67201494699264613970854563488, 1.40760963140939299703339722853, 3.19889084983765360749122782280, 3.66909026815544875517417204865, 4.61421178301317730203674562429, 6.32347291094455697687068877119, 6.87805639068679004354365406328, 8.853603793489960740004009032637, 9.549717271353841523012332278546, 9.913853741140368534270363753025, 10.74417178295774545564921316140

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