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Properties

Label 2-475-19.4-c1-0-6
Degree $2$
Conductor $475$
Sign $0.954 - 0.299i$
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  + (−2.18 − 0.795i)2-s + (0.199 + 1.13i)3-s + (2.60 + 2.18i)4-s + (0.463 − 2.62i)6-s + (0.0716 − 0.124i)7-s + (−1.63 − 2.82i)8-s + (1.57 − 0.574i)9-s + (1.40 + 2.43i)11-s + (−1.95 + 3.38i)12-s + (0.296 − 1.68i)13-s + (−0.255 + 0.214i)14-s + (0.135 + 0.766i)16-s + (−3.34 − 1.21i)17-s − 3.90·18-s + (2.82 − 3.32i)19-s + ⋯
L(s)  = 1  + (−1.54 − 0.562i)2-s + (0.115 + 0.653i)3-s + (1.30 + 1.09i)4-s + (0.189 − 1.07i)6-s + (0.0270 − 0.0469i)7-s + (−0.576 − 0.999i)8-s + (0.526 − 0.191i)9-s + (0.424 + 0.735i)11-s + (−0.564 + 0.977i)12-s + (0.0822 − 0.466i)13-s + (−0.0682 + 0.0572i)14-s + (0.0338 + 0.191i)16-s + (−0.811 − 0.295i)17-s − 0.920·18-s + (0.647 − 0.762i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.954 - 0.299i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.954 - 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.736098 + 0.112900i\)
\(L(\frac12)\) \(\approx\) \(0.736098 + 0.112900i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.82 + 3.32i)T \)
good2 \( 1 + (2.18 + 0.795i)T + (1.53 + 1.28i)T^{2} \)
3 \( 1 + (-0.199 - 1.13i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-0.0716 + 0.124i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.40 - 2.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.296 + 1.68i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (3.34 + 1.21i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-5.51 - 4.62i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (7.11 - 2.58i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.42 - 4.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.49T + 37T^{2} \)
41 \( 1 + (0.0325 + 0.184i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-8.42 + 7.06i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-3.44 + 1.25i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-1.25 - 1.05i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-5.04 - 1.83i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-7.74 - 6.50i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (5.64 - 2.05i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-0.414 + 0.347i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.81 - 10.3i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.01 + 11.4i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (3.79 - 6.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.251 + 1.42i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (8.27 + 3.01i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.86778715095494457897583177757, −10.00107846939539888481219858100, −9.203091023760918799183923640855, −8.960012545125784801905051976329, −7.40838167663131183086756362612, −7.09352230819591799793460764295, −5.26685971494050379030592980074, −4.00150760784360041686743489321, −2.68322000531649303138622631918, −1.20080759674952952106782376282, 0.942121306203171581841570501276, 2.19396436528057216671706506060, 4.13669356673589518301034534676, 5.89163344130734697436416724815, 6.68270485380787480312559923875, 7.48939945497549613209998436368, 8.204109967057641004253302561897, 9.088926856084681708893354915607, 9.741127580036743883317227204800, 10.86430208680862817987899091214

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