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Properties

Label 2-475-475.111-c1-0-4
Degree $2$
Conductor $475$
Sign $0.951 - 0.308i$
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  + (−0.920 − 0.0643i)2-s + (−0.115 − 3.31i)3-s + (−1.13 − 0.159i)4-s + (−0.216 + 2.22i)5-s + (−0.106 + 3.06i)6-s + (0.0485 + 0.0841i)7-s + (2.84 + 0.604i)8-s + (−7.99 + 0.559i)9-s + (0.342 − 2.03i)10-s + (0.398 + 3.79i)11-s + (−0.398 + 3.78i)12-s + (3.12 + 4.62i)13-s + (−0.0393 − 0.0805i)14-s + (7.40 + 0.460i)15-s + (−0.370 − 0.106i)16-s + (1.08 − 2.68i)17-s + ⋯
L(s)  = 1  + (−0.651 − 0.0455i)2-s + (−0.0668 − 1.91i)3-s + (−0.568 − 0.0798i)4-s + (−0.0968 + 0.995i)5-s + (−0.0436 + 1.24i)6-s + (0.0183 + 0.0317i)7-s + (1.00 + 0.213i)8-s + (−2.66 + 0.186i)9-s + (0.108 − 0.643i)10-s + (0.120 + 1.14i)11-s + (−0.114 + 1.09i)12-s + (0.865 + 1.28i)13-s + (−0.0105 − 0.0215i)14-s + (1.91 + 0.118i)15-s + (−0.0926 − 0.0265i)16-s + (0.263 − 0.652i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.951 - 0.308i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.951 - 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.616916 + 0.0975813i\)
\(L(\frac12)\) \(\approx\) \(0.616916 + 0.0975813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.216 - 2.22i)T \)
19 \( 1 + (0.378 - 4.34i)T \)
good2 \( 1 + (0.920 + 0.0643i)T + (1.98 + 0.278i)T^{2} \)
3 \( 1 + (0.115 + 3.31i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (-0.0485 - 0.0841i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.398 - 3.79i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (-3.12 - 4.62i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (-1.08 + 2.68i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (2.43 + 2.35i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (1.96 + 4.86i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (0.160 - 0.177i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-2.57 - 1.86i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.12 - 1.75i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (1.77 - 10.0i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.99 - 7.41i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (1.90 + 0.267i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (-2.71 - 10.8i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (-1.72 - 1.66i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (-9.44 - 5.90i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (8.60 + 4.57i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (-1.45 + 2.15i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (0.0861 + 2.46i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (6.34 - 7.04i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (11.5 - 3.29i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (-10.2 + 6.40i)T + (42.5 - 87.1i)T^{2} \)
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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

−11.26288970588989626507706750484, −10.07650681895183152879432790769, −9.138286590896817045480204726888, −8.058122386927565356383702283703, −7.51795601039468419748539797535, −6.67724880006784739730001710388, −5.89258452950558588886061842900, −4.17007968180006531669324318687, −2.40880370669848545909119680572, −1.39142524777356414067684831151, 0.53828720200771286182351869821, 3.45416325037047161524731218780, 4.08176366400988463833763454709, 5.24386602651158627913861668461, 5.73166390476701385484496836891, 7.975205906277881140793853457497, 8.687461884294273712769299132575, 9.035344487661790225149064066762, 9.987249968435823482081389014922, 10.71357709065526861864566420628

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