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Properties

Label 2-475-19.11-c1-0-18
Degree $2$
Conductor $475$
Sign $0.983 + 0.178i$
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  + (−0.155 + 0.269i)2-s + (1.12 − 1.94i)3-s + (0.951 + 1.64i)4-s + (0.349 + 0.604i)6-s + 3.96·7-s − 1.21·8-s + (−1.01 − 1.76i)9-s + 0.361·11-s + 4.27·12-s + (−1.25 − 2.17i)13-s + (−0.617 + 1.06i)14-s + (−1.71 + 2.96i)16-s + (−0.00464 + 0.00803i)17-s + 0.633·18-s + (−3.45 + 2.65i)19-s + ⋯
L(s)  = 1  + (−0.109 + 0.190i)2-s + (0.647 − 1.12i)3-s + (0.475 + 0.824i)4-s + (0.142 + 0.246i)6-s + 1.50·7-s − 0.429·8-s + (−0.339 − 0.587i)9-s + 0.108·11-s + 1.23·12-s + (−0.348 − 0.604i)13-s + (−0.165 + 0.285i)14-s + (−0.428 + 0.742i)16-s + (−0.00112 + 0.00194i)17-s + 0.149·18-s + (−0.793 + 0.609i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.983 + 0.178i$
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: \((2,\ 475,\ (\ :1/2),\ 0.983 + 0.178i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01350 - 0.181033i\)
\(L(\frac12)\) \(\approx\) \(2.01350 - 0.181033i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.45 - 2.65i)T \)
good2 \( 1 + (0.155 - 0.269i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.12 + 1.94i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 3.96T + 7T^{2} \)
11 \( 1 - 0.361T + 11T^{2} \)
13 \( 1 + (1.25 + 2.17i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.00464 - 0.00803i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.70 - 4.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.72 + 8.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.66T + 31T^{2} \)
37 \( 1 - 0.0596T + 37T^{2} \)
41 \( 1 + (1.85 - 3.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.10 + 3.64i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.45 + 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.48 + 9.50i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.65 + 4.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.44 - 7.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.32 - 4.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.68 - 13.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.83 - 8.36i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.70 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + (-2.08 - 3.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.87 + 3.24i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.42083523459798674206609504175, −10.08286341528978993546592449844, −8.577676998668760482851670383408, −8.157769479605353385178716237274, −7.53849097682163694850012135350, −6.75146875870722341184452330710, −5.43490807777781566439265872357, −3.99526827464811868725654850505, −2.58087012120811510063666994177, −1.66171642933155908020893713013, 1.64745820740410844217129635511, 2.89564707018477251374468674271, 4.51965393208119485609087909429, 4.90567731615423234223719183663, 6.33705869592074149550969234813, 7.48979234568587720470428849406, 8.750844420032352301168763966750, 9.174608944156344960858333621352, 10.27363624089657501190039413962, 10.89308570719035888725585007674

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