Properties

Label 2-475-19.11-c1-0-2
Degree $2$
Conductor $475$
Sign $-0.717 + 0.696i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.832 + 1.44i)2-s + (−0.579 + 1.00i)3-s + (−0.385 − 0.667i)4-s + (−0.965 − 1.67i)6-s + 2.43·7-s − 2.04·8-s + (0.827 + 1.43i)9-s − 5.75·11-s + 0.893·12-s + (−0.797 − 1.38i)13-s + (−2.02 + 3.51i)14-s + (2.47 − 4.28i)16-s + (−2.99 + 5.18i)17-s − 2.75·18-s + (0.149 + 4.35i)19-s + ⋯
L(s)  = 1  + (−0.588 + 1.01i)2-s + (−0.334 + 0.579i)3-s + (−0.192 − 0.333i)4-s + (−0.394 − 0.682i)6-s + 0.920·7-s − 0.723·8-s + (0.275 + 0.477i)9-s − 1.73·11-s + 0.258·12-s + (−0.221 − 0.383i)13-s + (−0.541 + 0.938i)14-s + (0.618 − 1.07i)16-s + (−0.725 + 1.25i)17-s − 0.649·18-s + (0.0342 + 0.999i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.717 + 0.696i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.717 + 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211714 - 0.521914i\)
\(L(\frac12)\) \(\approx\) \(0.211714 - 0.521914i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.149 - 4.35i)T \)
good2 \( 1 + (0.832 - 1.44i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.579 - 1.00i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 2.43T + 7T^{2} \)
11 \( 1 + 5.75T + 11T^{2} \)
13 \( 1 + (0.797 + 1.38i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.99 - 5.18i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.470 + 0.814i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.30 + 2.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 - 2.89T + 37T^{2} \)
41 \( 1 + (-3.15 + 5.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.26 + 3.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.47 - 7.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.09 + 1.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.39 + 9.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.26 - 9.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.504 - 0.874i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.41 - 7.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.12 - 8.87i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.80 - 6.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.11T + 83T^{2} \)
89 \( 1 + (-5.55 - 9.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.02 + 3.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.17899199114698021413062968229, −10.58170402805179353459346262338, −9.803092386344888153594438443712, −8.495773705185641376200674157386, −7.954785316602310433947704628783, −7.30819307433730056202549933353, −5.81862575174831443444016050713, −5.30024857304446586812343953342, −4.07715839251525170412492422850, −2.29323508449266073406875359089, 0.40962720218484536868611886743, 1.90074421752288511967960047843, 2.88674073307965261825454948967, 4.64892352457149715638169256068, 5.65570197556066702788076549681, 6.96345853090113631556430199140, 7.75153180958367891387072729062, 8.913112503613673153515157381568, 9.638324526639629460312854217087, 10.69525300022404574417665795087

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