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Properties

Label 2-475-95.18-c1-0-1
Degree $2$
Conductor $475$
Sign $0.535 - 0.844i$
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  + (−1.61 − 1.61i)2-s + (1.11 − 1.11i)3-s + 3.21i·4-s − 3.59·6-s + (−1.21 + 1.21i)7-s + (1.96 − 1.96i)8-s + 0.525i·9-s − 3.52·11-s + (3.57 + 3.57i)12-s + (−1.11 + 1.11i)13-s + 3.92·14-s + 0.0967·16-s + (−3.90 + 3.90i)17-s + (0.848 − 0.848i)18-s + (−3.92 + 1.90i)19-s + ⋯
L(s)  = 1  + (−1.14 − 1.14i)2-s + (0.642 − 0.642i)3-s + 1.60i·4-s − 1.46·6-s + (−0.458 + 0.458i)7-s + (0.693 − 0.693i)8-s + 0.175i·9-s − 1.06·11-s + (1.03 + 1.03i)12-s + (−0.308 + 0.308i)13-s + 1.04·14-s + 0.0241·16-s + (−0.946 + 0.946i)17-s + (0.199 − 0.199i)18-s + (−0.899 + 0.436i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.535 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242302 + 0.133224i\)
\(L(\frac12)\) \(\approx\) \(0.242302 + 0.133224i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.92 - 1.90i)T \)
good2 \( 1 + (1.61 + 1.61i)T + 2iT^{2} \)
3 \( 1 + (-1.11 + 1.11i)T - 3iT^{2} \)
7 \( 1 + (1.21 - 1.21i)T - 7iT^{2} \)
11 \( 1 + 3.52T + 11T^{2} \)
13 \( 1 + (1.11 - 1.11i)T - 13iT^{2} \)
17 \( 1 + (3.90 - 3.90i)T - 17iT^{2} \)
23 \( 1 + (1.21 + 1.21i)T + 23iT^{2} \)
29 \( 1 + 8.68T + 29T^{2} \)
31 \( 1 - 5.14iT - 31T^{2} \)
37 \( 1 + (2.11 + 2.11i)T + 37iT^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 + (-1.96 - 1.96i)T + 43iT^{2} \)
47 \( 1 + (-5.83 + 5.83i)T - 47iT^{2} \)
53 \( 1 + (0.107 - 0.107i)T - 53iT^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 9.13T + 61T^{2} \)
67 \( 1 + (6.56 + 6.56i)T + 67iT^{2} \)
71 \( 1 - 2.70iT - 71T^{2} \)
73 \( 1 + (4.65 + 4.65i)T + 73iT^{2} \)
79 \( 1 - 3.54T + 79T^{2} \)
83 \( 1 + (-3.06 - 3.06i)T + 83iT^{2} \)
89 \( 1 + 6.24T + 89T^{2} \)
97 \( 1 + (-1.42 - 1.42i)T + 97iT^{2} \)
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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

−10.71360254375725679025501362795, −10.51122551672391405699539876102, −9.234078322281199074564173484137, −8.634309902597369657555586751825, −7.936829765394761419737082054035, −6.98137604972612354927252139901, −5.55108677893998893132558198358, −3.80101082139876084236944909604, −2.45198009462632361819358735052, −1.96928119623476520534110042526, 0.20801180096831809538978631265, 2.68620898625272274669700106703, 4.11617333241439749602750608160, 5.41507263895034297546738696415, 6.55734294364443853425247154213, 7.36892467100228858636065242100, 8.208609704996839924451922319287, 9.056197931462991781845469098595, 9.701204072359269703153620587606, 10.31881613453358900164835744611

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