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Properties

Label 2-475-19.7-c1-0-17
Degree $2$
Conductor $475$
Sign $0.973 + 0.230i$
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  + (0.548 + 0.950i)2-s + (−0.189 − 0.328i)3-s + (0.397 − 0.689i)4-s + (0.208 − 0.360i)6-s − 1.89·7-s + 3.06·8-s + (1.42 − 2.47i)9-s + 0.134·11-s − 0.301·12-s + (1.75 − 3.04i)13-s + (−1.03 − 1.79i)14-s + (0.887 + 1.53i)16-s + (−0.830 − 1.43i)17-s + 3.13·18-s + (2.10 + 3.81i)19-s + ⋯
L(s)  = 1  + (0.388 + 0.672i)2-s + (−0.109 − 0.189i)3-s + (0.198 − 0.344i)4-s + (0.0849 − 0.147i)6-s − 0.715·7-s + 1.08·8-s + (0.476 − 0.824i)9-s + 0.0405·11-s − 0.0871·12-s + (0.487 − 0.843i)13-s + (−0.277 − 0.480i)14-s + (0.221 + 0.384i)16-s + (−0.201 − 0.348i)17-s + 0.738·18-s + (0.483 + 0.875i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.973 + 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80972 - 0.211070i\)
\(L(\frac12)\) \(\approx\) \(1.80972 - 0.211070i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 - 3.81i)T \)
good2 \( 1 + (-0.548 - 0.950i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.189 + 0.328i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 1.89T + 7T^{2} \)
11 \( 1 - 0.134T + 11T^{2} \)
13 \( 1 + (-1.75 + 3.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.830 + 1.43i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.68 + 4.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.48 - 4.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.56T + 31T^{2} \)
37 \( 1 - 1.69T + 37T^{2} \)
41 \( 1 + (5.31 + 9.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.25 - 7.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.55 - 9.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.132 - 0.229i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.44 - 5.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.58 - 7.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.47 - 2.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.664 + 1.15i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.17 + 5.49i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.733 - 1.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.44T + 83T^{2} \)
89 \( 1 + (4.86 - 8.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.73 - 15.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.84498149652409040364161625529, −10.12143734502893398526232081148, −9.254052007204420380442960353810, −8.008031605851936907439219032852, −7.01307831494526134922718117752, −6.34235183222233604465242411324, −5.57409346539909323179645109949, −4.34324464075576550421216570738, −3.07063105692027031916533366581, −1.14554388903149267243860637038, 1.79002817981708137944464928823, 3.06950599784588726674612694061, 4.09716421848123618989328297035, 5.03130422040475362489046964269, 6.48798317895287024766365830741, 7.31174517680356638111678941233, 8.329907342349035346752816377368, 9.532228125307534989213498712654, 10.26763500814723589783017970048, 11.33702879425274948380550966579

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