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Properties

Label 2-475-95.44-c1-0-12
Degree $2$
Conductor $475$
Sign $0.0199 - 0.999i$
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  + (1.16 + 1.39i)2-s + (−0.0605 + 0.166i)3-s + (−0.226 + 1.28i)4-s + (−0.302 + 0.110i)6-s + (−0.929 − 0.536i)7-s + (1.09 − 0.634i)8-s + (2.27 + 1.90i)9-s + (1.65 + 2.86i)11-s + (−0.199 − 0.115i)12-s + (0.908 + 2.49i)13-s + (−0.338 − 1.92i)14-s + (4.61 + 1.67i)16-s + (−2.57 − 3.06i)17-s + 5.39i·18-s + (−0.281 + 4.34i)19-s + ⋯
L(s)  = 1  + (0.825 + 0.984i)2-s + (−0.0349 + 0.0960i)3-s + (−0.113 + 0.640i)4-s + (−0.123 + 0.0449i)6-s + (−0.351 − 0.202i)7-s + (0.388 − 0.224i)8-s + (0.758 + 0.636i)9-s + (0.499 + 0.864i)11-s + (−0.0576 − 0.0332i)12-s + (0.252 + 0.692i)13-s + (−0.0905 − 0.513i)14-s + (1.15 + 0.419i)16-s + (−0.623 − 0.743i)17-s + 1.27i·18-s + (−0.0646 + 0.997i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0199 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0199 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.0199 - 0.999i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.0199 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61881 + 1.58687i\)
\(L(\frac12)\) \(\approx\) \(1.61881 + 1.58687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.281 - 4.34i)T \)
good2 \( 1 + (-1.16 - 1.39i)T + (-0.347 + 1.96i)T^{2} \)
3 \( 1 + (0.0605 - 0.166i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (0.929 + 0.536i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.65 - 2.86i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.908 - 2.49i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.57 + 3.06i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.72 + 0.304i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (1.72 + 1.44i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-4.02 + 6.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.64iT - 37T^{2} \)
41 \( 1 + (-0.842 - 0.306i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (8.38 - 1.47i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-4.04 + 4.82i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (-3.34 - 0.590i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-1.13 + 0.955i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.38 + 13.5i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (8.26 - 9.85i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.91 - 10.8i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-0.892 + 2.45i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (6.17 + 2.24i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.33 - 1.34i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.742 + 0.270i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (12.0 + 14.3i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.33177350858110823703059681905, −10.13667950190294712527908151421, −9.613364086039419595078671083096, −8.187283424304901652358926349776, −7.20045098895738868206837230040, −6.68067150153149674942796065107, −5.61068169889473575004769044842, −4.48682747116290416011107432366, −3.95307642137651750980476562793, −1.89056883058898781157591675127, 1.30879954665644084280549293449, 2.88570573880855484445958846087, 3.72585035584499886928026728822, 4.72779388296933398015236188322, 5.97738876145436670913623152649, 6.87542512281416962546565929562, 8.202103838443601925018218023284, 9.131977492463483352329876935175, 10.26279331843981509845137559972, 10.91215139731950342406187369090

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