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Properties

Label 2-475-95.74-c1-0-17
Degree $2$
Conductor $475$
Sign $0.599 - 0.800i$
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 + 0.385i)2-s + (−0.666 + 0.794i)3-s + (2.76 + 1.00i)4-s + (−1.76 + 1.48i)6-s + (1.75 − 1.01i)7-s + (1.81 + 1.04i)8-s + (0.334 + 1.89i)9-s + (−0.0424 + 0.0734i)11-s + (−2.64 + 1.52i)12-s + (3.67 + 4.38i)13-s + (4.22 − 1.53i)14-s + (−0.945 − 0.793i)16-s + (2.49 + 0.439i)17-s + 4.27i·18-s + (−3.21 − 2.94i)19-s + ⋯
L(s)  = 1  + (1.54 + 0.272i)2-s + (−0.384 + 0.458i)3-s + (1.38 + 0.502i)4-s + (−0.720 + 0.604i)6-s + (0.662 − 0.382i)7-s + (0.640 + 0.369i)8-s + (0.111 + 0.631i)9-s + (−0.0127 + 0.0221i)11-s + (−0.762 + 0.440i)12-s + (1.01 + 1.21i)13-s + (1.13 − 0.411i)14-s + (−0.236 − 0.198i)16-s + (0.604 + 0.106i)17-s + 1.00i·18-s + (−0.737 − 0.675i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.599 - 0.800i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.599 - 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68983 + 1.34688i\)
\(L(\frac12)\) \(\approx\) \(2.68983 + 1.34688i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.21 + 2.94i)T \)
good2 \( 1 + (-2.18 - 0.385i)T + (1.87 + 0.684i)T^{2} \)
3 \( 1 + (0.666 - 0.794i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (-1.75 + 1.01i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.0424 - 0.0734i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.67 - 4.38i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.49 - 0.439i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (0.105 - 0.290i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.455 + 2.58i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (4.03 + 6.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.01iT - 37T^{2} \)
41 \( 1 + (4.50 + 3.77i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.222 + 0.611i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-6.79 + 1.19i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-4.97 + 13.6i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.29 - 7.35i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (12.5 + 4.55i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-8.74 + 1.54i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-13.4 + 4.90i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (6.97 - 8.31i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (0.0887 + 0.0744i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (2.57 - 1.48i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.52 + 7.15i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.222 - 0.0392i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.30806976962792616336040760069, −10.70747558011289016635689724245, −9.453893398080564902234116343882, −8.218883895738163525132077007605, −7.13276669318231050058793341542, −6.18937056171279327299404517438, −5.26754629565226497130140522680, −4.41805637016388621976448439190, −3.78653092866955195887282657865, −2.08212018407623997409978894408, 1.51821123644605632181284651767, 3.09396120805113129533099282750, 4.00074857017344796119686100968, 5.33429320885779357810700661483, 5.82863296344545450276741433854, 6.77033475368768989543549675264, 8.001673213274638057619169874231, 9.002623392804338999006969024821, 10.55583816133968262364539969142, 11.10474339714940608271891568050

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