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Properties

Label 2-475-19.7-c1-0-21
Degree $2$
Conductor $475$
Sign $-0.0977 + 0.995i$
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 − 1.73i)3-s + (1 − 1.73i)4-s + 4·7-s + (−0.499 + 0.866i)9-s + 3·11-s − 3.99·12-s + (1 − 1.73i)13-s + (−1.99 − 3.46i)16-s + (3 + 5.19i)17-s + (−3.5 + 2.59i)19-s + (−4 − 6.92i)21-s − 4.00·27-s + (4 − 6.92i)28-s + (1.5 − 2.59i)29-s − 7·31-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + (0.5 − 0.866i)4-s + 1.51·7-s + (−0.166 + 0.288i)9-s + 0.904·11-s − 1.15·12-s + (0.277 − 0.480i)13-s + (−0.499 − 0.866i)16-s + (0.727 + 1.26i)17-s + (−0.802 + 0.596i)19-s + (−0.872 − 1.51i)21-s − 0.769·27-s + (0.755 − 1.30i)28-s + (0.278 − 0.482i)29-s − 1.25·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.0977 + 0.995i$
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: \((2,\ 475,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06118 - 1.17050i\)
\(L(\frac12)\) \(\approx\) \(1.06118 - 1.17050i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 2.59i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.5 - 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4 - 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4 - 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.86667897450936964890459142585, −10.23460724456847848843181183484, −8.812319320166885670974834814309, −7.88098328543309962338861424988, −6.99493743745264066898582476958, −6.03635348207729154368808649323, −5.45898555686003375075929085438, −4.03663261716282539335541559312, −1.86919817563922856821254525608, −1.24096142123401968690163984206, 1.90450901963427336876409659814, 3.59627781104575035677850307993, 4.53081813332288507663557053442, 5.30231486765980986190015992088, 6.72053709103147934181714728347, 7.60332834659405498782401041705, 8.614350793160160309967491943006, 9.412174090981647598879264232348, 10.71593279964040892144337013764, 11.25680112181617328687651782649

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