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Properties

Label 2-475-475.56-c0-0-0
Degree $2$
Conductor $475$
Sign $0.968 + 0.248i$
Analytic cond. $0.237055$
Root an. cond. $0.486883$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + 0.618·7-s + (0.309 − 0.951i)9-s + (0.190 + 0.587i)11-s + (0.309 − 0.951i)16-s + (1.30 + 0.951i)17-s + (−0.809 − 0.587i)19-s + (0.309 + 0.951i)20-s + (0.190 + 0.587i)23-s + (−0.809 − 0.587i)25-s + (−0.500 + 0.363i)28-s + (0.190 − 0.587i)35-s + (0.309 + 0.951i)36-s − 1.61·43-s + (−0.5 − 0.363i)44-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + 0.618·7-s + (0.309 − 0.951i)9-s + (0.190 + 0.587i)11-s + (0.309 − 0.951i)16-s + (1.30 + 0.951i)17-s + (−0.809 − 0.587i)19-s + (0.309 + 0.951i)20-s + (0.190 + 0.587i)23-s + (−0.809 − 0.587i)25-s + (−0.500 + 0.363i)28-s + (0.190 − 0.587i)35-s + (0.309 + 0.951i)36-s − 1.61·43-s + (−0.5 − 0.363i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(0.237055\)
Root analytic conductor: \(0.486883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :0),\ 0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8385586567\)
\(L(\frac12)\) \(\approx\) \(0.8385586567\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (0.809 - 0.587i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 - 0.618T + T^{2} \)
11 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−11.37886454512092168024259263242, −9.910558876033988650893968898905, −9.448762793123509315103132686116, −8.452419387931001697072544752155, −7.888705089503700860205246866215, −6.55089706267017693924717340692, −5.27836402058779652395305510662, −4.47260810060413977729218084541, −3.50517554469108556152813166591, −1.46582343848180890479956299178, 1.77277782451024716362909059922, 3.34161278065914401490037529893, 4.71224558264376277973019136988, 5.51754326709316008496193761884, 6.55261603617343811591781833819, 7.75199927279110144898271175156, 8.501925301299130963833091403594, 9.726106130176514588999452767824, 10.34057463346922722045445371387, 11.01240611397593807139001346037

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