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Properties

Label 2-475-19.7-c1-0-6
Degree $2$
Conductor $475$
Sign $0.971 - 0.235i$
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  + (−0.595 − 1.03i)2-s + (1.52 + 2.63i)3-s + (0.290 − 0.503i)4-s + (1.81 − 3.14i)6-s + 0.609·7-s − 3.07·8-s + (−3.14 + 5.44i)9-s + 4.48·11-s + 1.77·12-s + (2.21 − 3.84i)13-s + (−0.362 − 0.628i)14-s + (1.24 + 2.16i)16-s + (1.45 + 2.51i)17-s + 7.48·18-s + (3.60 + 2.44i)19-s + ⋯
L(s)  = 1  + (−0.421 − 0.729i)2-s + (0.879 + 1.52i)3-s + (0.145 − 0.251i)4-s + (0.740 − 1.28i)6-s + 0.230·7-s − 1.08·8-s + (−1.04 + 1.81i)9-s + 1.35·11-s + 0.511·12-s + (0.615 − 1.06i)13-s + (−0.0969 − 0.167i)14-s + (0.312 + 0.540i)16-s + (0.352 + 0.609i)17-s + 1.76·18-s + (0.827 + 0.562i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.971 - 0.235i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.971 - 0.235i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67156 + 0.199441i\)
\(L(\frac12)\) \(\approx\) \(1.67156 + 0.199441i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 - 2.44i)T \)
good2 \( 1 + (0.595 + 1.03i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.52 - 2.63i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 0.609T + 7T^{2} \)
11 \( 1 - 4.48T + 11T^{2} \)
13 \( 1 + (-2.21 + 3.84i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.45 - 2.51i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.42 - 2.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.558 - 0.966i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.22T + 31T^{2} \)
37 \( 1 - 3.77T + 37T^{2} \)
41 \( 1 + (-4.15 - 7.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.99 + 8.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.94 - 5.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.22 + 7.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.11 + 8.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.49 + 4.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.23 + 7.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.80 + 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.86 - 3.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.51 + 7.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.12T + 83T^{2} \)
89 \( 1 + (3.96 - 6.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.83 + 8.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.90816974921579773340553938152, −9.956019079072939973617940596047, −9.559397712025654047492132786780, −8.722526392021306329443494033133, −7.896130240371761615714254614820, −6.15618941897532590999549231599, −5.18466572154839678082627803204, −3.73733934205580165168866284635, −3.24966801356794041294496311797, −1.63696031685538137958470236119, 1.32441814633975161070685139838, 2.67515084037100243000498400312, 3.87643831528172820034544591919, 5.92418117004596319141726006682, 6.80784026856440104731748521757, 7.22250633010283982380809716694, 8.135261508518811182522790357618, 8.956839352966792652718048058152, 9.381260468892437955985923750934, 11.50318777900986501428568226908

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