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Properties

Label 2-475-95.54-c1-0-12
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: $\chi_{475} (149, \cdot )$
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

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

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