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Properties

Label 2-475-19.18-c2-0-18
Degree $2$
Conductor $475$
Sign $0.966 + 0.255i$
Analytic cond. $12.9428$
Root an. cond. $3.59761$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.01i·2-s + 4.83i·3-s − 5.10·4-s + 14.5·6-s + 6.15·7-s + 3.33i·8-s − 14.3·9-s + 5.09·11-s − 24.6i·12-s − 4.20i·13-s − 18.5i·14-s − 10.3·16-s − 1.29·17-s + 43.2i·18-s + (18.3 + 4.84i)19-s + ⋯
L(s)  = 1  − 1.50i·2-s + 1.61i·3-s − 1.27·4-s + 2.43·6-s + 0.879·7-s + 0.417i·8-s − 1.59·9-s + 0.463·11-s − 2.05i·12-s − 0.323i·13-s − 1.32i·14-s − 0.647·16-s − 0.0764·17-s + 2.40i·18-s + (0.966 + 0.255i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.966 + 0.255i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ 0.966 + 0.255i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.874767032\)
\(L(\frac12)\) \(\approx\) \(1.874767032\)
\(L(2)\) 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 \( 1 + (-18.3 - 4.84i)T \)
good2 \( 1 + 3.01iT - 4T^{2} \)
3 \( 1 - 4.83iT - 9T^{2} \)
7 \( 1 - 6.15T + 49T^{2} \)
11 \( 1 - 5.09T + 121T^{2} \)
13 \( 1 + 4.20iT - 169T^{2} \)
17 \( 1 + 1.29T + 289T^{2} \)
23 \( 1 - 37.1T + 529T^{2} \)
29 \( 1 - 42.9iT - 841T^{2} \)
31 \( 1 - 52.1iT - 961T^{2} \)
37 \( 1 + 41.8iT - 1.36e3T^{2} \)
41 \( 1 - 7.13iT - 1.68e3T^{2} \)
43 \( 1 - 59.2T + 1.84e3T^{2} \)
47 \( 1 - 57.1T + 2.20e3T^{2} \)
53 \( 1 + 61.6iT - 2.80e3T^{2} \)
59 \( 1 - 54.5iT - 3.48e3T^{2} \)
61 \( 1 - 38.4T + 3.72e3T^{2} \)
67 \( 1 - 44.7iT - 4.48e3T^{2} \)
71 \( 1 + 13.0iT - 5.04e3T^{2} \)
73 \( 1 + 134.T + 5.32e3T^{2} \)
79 \( 1 + 128. iT - 6.24e3T^{2} \)
83 \( 1 - 23.2T + 6.88e3T^{2} \)
89 \( 1 + 21.0iT - 7.92e3T^{2} \)
97 \( 1 - 75.9iT - 9.40e3T^{2} \)
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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

−10.71874681585856457600173360251, −10.24586949422651545261406458372, −9.075065220777082878745237308069, −8.893749968709971190189943888561, −7.22126320401414072911105960503, −5.36591930747285875521776872616, −4.70745190164418263781465386290, −3.70306237777174691119882414228, −2.91600496434150439386235936066, −1.24854426526986035397361185356, 0.959043253058744193689065296022, 2.40354132547143984378784536021, 4.49383025677784774560601458708, 5.63016197861543159007287784360, 6.38521883043223941633472655093, 7.29539071025123542721428986741, 7.69321493821279594573993945781, 8.549356228881377521377921194212, 9.414236368188495694316042272327, 11.28019311047555990352462208367

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