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Properties

Label 2-475-19.5-c1-0-23
Degree $2$
Conductor $475$
Sign $0.456 + 0.889i$
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  + (2.42 − 0.883i)2-s + (0.0430 − 0.243i)3-s + (3.57 − 3.00i)4-s + (−0.111 − 0.630i)6-s + (−0.200 − 0.347i)7-s + (3.45 − 5.97i)8-s + (2.76 + 1.00i)9-s + (−2.59 + 4.49i)11-s + (−0.578 − 1.00i)12-s + (−0.501 − 2.84i)13-s + (−0.794 − 0.666i)14-s + (1.47 − 8.35i)16-s + (−3.89 + 1.41i)17-s + 7.59·18-s + (0.386 − 4.34i)19-s + ⋯
L(s)  = 1  + (1.71 − 0.624i)2-s + (0.0248 − 0.140i)3-s + (1.78 − 1.50i)4-s + (−0.0453 − 0.257i)6-s + (−0.0759 − 0.131i)7-s + (1.22 − 2.11i)8-s + (0.920 + 0.335i)9-s + (−0.782 + 1.35i)11-s + (−0.167 − 0.289i)12-s + (−0.139 − 0.789i)13-s + (−0.212 − 0.178i)14-s + (0.368 − 2.08i)16-s + (−0.944 + 0.343i)17-s + 1.78·18-s + (0.0887 − 0.996i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.456 + 0.889i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.456 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.11233 - 1.90188i\)
\(L(\frac12)\) \(\approx\) \(3.11233 - 1.90188i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.386 + 4.34i)T \)
good2 \( 1 + (-2.42 + 0.883i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (-0.0430 + 0.243i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (0.200 + 0.347i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 - 4.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.501 + 2.84i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (3.89 - 1.41i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (2.57 - 2.15i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (6.18 + 2.25i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.13 - 5.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.14T + 37T^{2} \)
41 \( 1 + (-0.496 + 2.81i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-9.52 - 7.99i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-6.35 - 2.31i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (9.42 - 7.90i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-1.42 + 0.518i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (5.35 - 4.49i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.711 + 0.258i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (6.38 + 5.35i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.72 + 9.76i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.553 + 3.13i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.75 - 4.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.17 + 12.3i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (8.35 - 3.04i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.88550658018295391867234728188, −10.45211592318772714008081087028, −9.462223308160128351086135138746, −7.65465099306146274414216415542, −7.00031368976445030576971374739, −5.84566827361744806089999731898, −4.78220220917853426898179581617, −4.24766360627880129276782404195, −2.80935950597319151182313273784, −1.83058125435032782845902047937, 2.36470846400093948421602659457, 3.66241206166661316253845815353, 4.37681040396765721574754811964, 5.52452058718486033385498423286, 6.25846842633420978662407959755, 7.17892754111309789364343818637, 8.084775580595483723186874587765, 9.298241107501576076778925156801, 10.63400093238274031432356948829, 11.44168677893965484328179165466

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