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Properties

Label 2-475-95.49-c1-0-26
Degree $2$
Conductor $475$
Sign $-0.950 - 0.311i$
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  + (0.269 + 0.155i)2-s + (−0.891 − 0.514i)3-s + (−0.951 − 1.64i)4-s + (−0.160 − 0.277i)6-s + 3.28i·7-s − 1.21i·8-s + (−0.969 − 1.67i)9-s − 5.16·11-s + 1.95i·12-s + (3.06 − 1.76i)13-s + (−0.510 + 0.883i)14-s + (−1.71 + 2.96i)16-s + (−0.874 − 0.504i)17-s − 0.603i·18-s + (−2.42 + 3.62i)19-s + ⋯
L(s)  = 1  + (0.190 + 0.109i)2-s + (−0.514 − 0.297i)3-s + (−0.475 − 0.824i)4-s + (−0.0653 − 0.113i)6-s + 1.23i·7-s − 0.429i·8-s + (−0.323 − 0.559i)9-s − 1.55·11-s + 0.565i·12-s + (0.849 − 0.490i)13-s + (−0.136 + 0.236i)14-s + (−0.428 + 0.742i)16-s + (−0.211 − 0.122i)17-s − 0.142i·18-s + (−0.555 + 0.831i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.950 - 0.311i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.950 - 0.311i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0103990 + 0.0651140i\)
\(L(\frac12)\) \(\approx\) \(0.0103990 + 0.0651140i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.42 - 3.62i)T \)
good2 \( 1 + (-0.269 - 0.155i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.891 + 0.514i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 3.28iT - 7T^{2} \)
11 \( 1 + 5.16T + 11T^{2} \)
13 \( 1 + (-3.06 + 1.76i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.874 + 0.504i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (6.63 - 3.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.01 + 3.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.60T + 31T^{2} \)
37 \( 1 - 6.48iT - 37T^{2} \)
41 \( 1 + (-3.40 + 5.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.46 + 3.15i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.32 + 1.92i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.16 - 3.55i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.73 + 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.06 + 5.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.69 + 5.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.227 + 0.394i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.57 + 2.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.44 + 2.50i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.50iT - 83T^{2} \)
89 \( 1 + (3.56 + 6.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.37 + 5.41i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.54066200408440172721474831203, −9.681355970485950842822229924378, −8.717659021260646171862620580386, −7.907119031074305995769225560483, −6.33006058331029380728282262210, −5.74345084396075726433924440445, −5.20445122753549504488795319625, −3.60819383771314825700735508131, −2.00247008171763100767509889759, −0.03815518478951578280081739480, 2.51260470612312582947567855541, 3.94288792036145514134473985951, 4.62810042169334316423267044154, 5.68544420850676684501925294876, 7.04322617377357204805915703109, 7.944474362819047000955080718074, 8.613858103824497708540616003371, 9.938791602407527110984978917880, 10.90090705603377820722169560175, 11.13666535350970274640568970837

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