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Properties

Label 2-475-95.44-c1-0-4
Degree $2$
Conductor $475$
Sign $0.787 + 0.615i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.787 + 0.615i$
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: \((2,\ 475,\ (\ :1/2),\ 0.787 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.944198 - 0.325219i\)
\(L(\frac12)\) \(\approx\) \(0.944198 - 0.325219i\)
\(L(\frac{3}{2})\) 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

−10.75484233017926861452391655575, −9.980997757047284321603155065394, −9.537704180623692412700774377314, −8.183719059216834466821509870473, −7.74928119307850950301843392454, −6.29948146751700379084355519254, −5.10789727343954680730244012631, −3.81091641767339881475628124295, −2.33014378350600107345332342276, −1.35916899776531721358062996789, 0.965840531177652111333071454745, 3.17086892698223808566291133083, 4.46181576220446930255487931358, 5.76960955781277995356571665193, 6.88054396494467755615378005984, 7.25877812573898083817339188120, 8.483619035891380355422379735061, 9.149559160618285586454783990976, 9.808330512283311082921830039895, 10.97926349653450579615477235768

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