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Properties

Label 2-475-19.11-c1-0-16
Degree $2$
Conductor $475$
Sign $0.332 + 0.943i$
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  + (1.08 − 1.87i)2-s + (−0.706 + 1.22i)3-s + (−1.35 − 2.34i)4-s + (1.53 + 2.65i)6-s + 1.76·7-s − 1.53·8-s + (0.502 + 0.869i)9-s + 1.83·11-s + 3.82·12-s + (−1.30 − 2.25i)13-s + (1.91 − 3.31i)14-s + (1.03 − 1.79i)16-s + (2.11 − 3.66i)17-s + 2.17·18-s + (4.01 + 1.68i)19-s + ⋯
L(s)  = 1  + (0.767 − 1.32i)2-s + (−0.407 + 0.706i)3-s + (−0.677 − 1.17i)4-s + (0.625 + 1.08i)6-s + 0.665·7-s − 0.544·8-s + (0.167 + 0.289i)9-s + 0.554·11-s + 1.10·12-s + (−0.361 − 0.625i)13-s + (0.510 − 0.884i)14-s + (0.259 − 0.449i)16-s + (0.513 − 0.889i)17-s + 0.513·18-s + (0.922 + 0.386i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.332 + 0.943i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.332 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68206 - 1.19028i\)
\(L(\frac12)\) \(\approx\) \(1.68206 - 1.19028i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.01 - 1.68i)T \)
good2 \( 1 + (-1.08 + 1.87i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.706 - 1.22i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.76T + 7T^{2} \)
11 \( 1 - 1.83T + 11T^{2} \)
13 \( 1 + (1.30 + 2.25i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.11 + 3.66i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.10 + 1.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.56 - 6.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.303T + 31T^{2} \)
37 \( 1 + 3.90T + 37T^{2} \)
41 \( 1 + (4.11 - 7.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.17 + 2.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.62 + 6.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.31 - 9.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.02 + 10.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.26 + 9.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.51 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.91 - 10.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.58 - 7.94i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.94 - 6.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.93T + 83T^{2} \)
89 \( 1 + (-6.23 - 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.87 + 6.71i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.99989889606092394000058552805, −10.14442306589116038060584393285, −9.684881741262587195575539495634, −8.249870683064506534727862003919, −7.13148562562567071885866991758, −5.37830943326653019618461497525, −4.98331111268418229389663113995, −3.94647700606612471669230691532, −2.87291219605138483216826201347, −1.39060147000619639032962851981, 1.55624551876429758187879874596, 3.73892980984168203993003651787, 4.74526988216744419622245001623, 5.77177282737129438485780330387, 6.50441593343956871567148406917, 7.30562440121038654567632822860, 7.979300518497352476717768565332, 9.092044827077187251482868267804, 10.29104767220110386807721191703, 11.68950967558571513091075038475

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