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Properties

Label 2-475-19.11-c1-0-27
Degree $2$
Conductor $475$
Sign $-0.948 - 0.315i$
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  + (1.22 − 2.12i)2-s + (0.780 − 1.35i)3-s + (−2.01 − 3.49i)4-s + (−1.91 − 3.32i)6-s − 4.50·7-s − 4.99·8-s + (0.281 + 0.487i)9-s + 2.19·11-s − 6.29·12-s + (−1.87 − 3.25i)13-s + (−5.53 + 9.58i)14-s + (−2.09 + 3.63i)16-s + (0.332 − 0.576i)17-s + 1.38·18-s + (3.79 − 2.13i)19-s + ⋯
L(s)  = 1  + (0.868 − 1.50i)2-s + (0.450 − 0.780i)3-s + (−1.00 − 1.74i)4-s + (−0.782 − 1.35i)6-s − 1.70·7-s − 1.76·8-s + (0.0938 + 0.162i)9-s + 0.662·11-s − 1.81·12-s + (−0.521 − 0.902i)13-s + (−1.47 + 2.56i)14-s + (−0.524 + 0.909i)16-s + (0.0807 − 0.139i)17-s + 0.326·18-s + (0.871 − 0.490i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.948 - 0.315i$
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: \((2,\ 475,\ (\ :1/2),\ -0.948 - 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.320665 + 1.97788i\)
\(L(\frac12)\) \(\approx\) \(0.320665 + 1.97788i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.79 + 2.13i)T \)
good2 \( 1 + (-1.22 + 2.12i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.780 + 1.35i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 4.50T + 7T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 + (1.87 + 3.25i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.332 + 0.576i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.244 + 0.422i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.79 + 3.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.83T + 31T^{2} \)
37 \( 1 - 3.01T + 37T^{2} \)
41 \( 1 + (0.0362 - 0.0627i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.210 + 0.364i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.51 - 4.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.30 - 2.26i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.26 - 10.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.53 + 6.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.86 - 4.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.48 - 6.03i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.47 - 2.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.66 - 9.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 + (-0.668 - 1.15i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.19 + 3.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.54468651722662433462532225830, −9.850755407635462809250915872701, −9.184050510872669634407662979103, −7.71248591063186158214913780245, −6.66694026907708763176770367140, −5.61489937796486723114796045550, −4.31514187609089957593355676033, −3.11495598124505707662667730001, −2.57266278393050319686771701531, −0.934105710796841451014442690212, 3.21669337015762043844747614566, 3.85958298314530095996770122887, 4.82542413865052279038440116412, 6.13676523878285161541362112955, 6.65059445769926746956368275160, 7.54497861277769649577090389539, 8.876434038020790553539397147576, 9.437302261584886238329526965427, 10.18643917463953540024314812362, 11.94248107191329901240153117014

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