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Properties

Label 2-475-19.11-c1-0-26
Degree $2$
Conductor $475$
Sign $-0.989 - 0.143i$
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  + (1.37 − 2.38i)2-s + (0.745 − 1.29i)3-s + (−2.80 − 4.85i)4-s + (−2.05 − 3.56i)6-s + 2.84·7-s − 9.94·8-s + (0.387 + 0.670i)9-s − 0.864·11-s − 8.36·12-s + (0.321 + 0.557i)13-s + (3.92 − 6.80i)14-s + (−8.11 + 14.0i)16-s + (1.87 − 3.24i)17-s + 2.13·18-s + (−3.36 + 2.77i)19-s + ⋯
L(s)  = 1  + (0.975 − 1.68i)2-s + (0.430 − 0.745i)3-s + (−1.40 − 2.42i)4-s + (−0.839 − 1.45i)6-s + 1.07·7-s − 3.51·8-s + (0.129 + 0.223i)9-s − 0.260·11-s − 2.41·12-s + (0.0892 + 0.154i)13-s + (1.04 − 1.81i)14-s + (−2.02 + 3.51i)16-s + (0.453 − 0.785i)17-s + 0.503·18-s + (−0.770 + 0.636i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.989 - 0.143i$
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: \((2,\ 475,\ (\ :1/2),\ -0.989 - 0.143i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.182467 + 2.53073i\)
\(L(\frac12)\) \(\approx\) \(0.182467 + 2.53073i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.36 - 2.77i)T \)
good2 \( 1 + (-1.37 + 2.38i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.745 + 1.29i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 2.84T + 7T^{2} \)
11 \( 1 + 0.864T + 11T^{2} \)
13 \( 1 + (-0.321 - 0.557i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.87 + 3.24i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.208 - 0.361i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.85 - 8.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.93T + 31T^{2} \)
37 \( 1 + 6.36T + 37T^{2} \)
41 \( 1 + (-2.00 + 3.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.02 - 1.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.97 + 3.42i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.49 + 9.51i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.22 - 2.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.16 + 5.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.26 + 2.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.891 + 1.54i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.56 - 6.17i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.912 - 1.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.43T + 83T^{2} \)
89 \( 1 + (2.22 + 3.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.42 - 9.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.72594481772830030218598826838, −10.12489801073201901230512390366, −8.877667489640092717035089331127, −8.048120363819779116714166283563, −6.65065579405777033372038718522, −5.23718591683612163468279329490, −4.64710548838404934511461104819, −3.30192200281287034265994479783, −2.18149085859799448519180150318, −1.31853484710184866130905223205, 3.01149360063425904368775758682, 4.27546603225318642787954451264, 4.66645692282631909992835990889, 5.84648111776949159182715436844, 6.72235115742122324169428922760, 7.935206559866611080412342736790, 8.350996517966256633347288828110, 9.271345510582714791187582058122, 10.46663614701520367970420793168, 11.79654071547316599898606731889

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