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Properties

Label 2-475-19.18-c2-0-52
Degree $2$
Conductor $475$
Sign $-0.601 - 0.798i$
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  − 2.92i·2-s − 2.02i·3-s − 4.57·4-s − 5.92·6-s + 8.32·7-s + 1.68i·8-s + 4.90·9-s − 17.3·11-s + 9.25i·12-s − 5.27i·13-s − 24.3i·14-s − 13.3·16-s − 19.0·17-s − 14.3i·18-s + (−11.4 − 15.1i)19-s + ⋯
L(s)  = 1  − 1.46i·2-s − 0.674i·3-s − 1.14·4-s − 0.987·6-s + 1.18·7-s + 0.211i·8-s + 0.545·9-s − 1.57·11-s + 0.771i·12-s − 0.405i·13-s − 1.74i·14-s − 0.834·16-s − 1.12·17-s − 0.798i·18-s + (−0.601 − 0.798i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.601 - 0.798i$
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: \((2,\ 475,\ (\ :1),\ -0.601 - 0.798i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.351372004\)
\(L(\frac12)\) \(\approx\) \(1.351372004\)
\(L(2)\) 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 \( 1 + (11.4 + 15.1i)T \)
good2 \( 1 + 2.92iT - 4T^{2} \)
3 \( 1 + 2.02iT - 9T^{2} \)
7 \( 1 - 8.32T + 49T^{2} \)
11 \( 1 + 17.3T + 121T^{2} \)
13 \( 1 + 5.27iT - 169T^{2} \)
17 \( 1 + 19.0T + 289T^{2} \)
23 \( 1 + 3.32T + 529T^{2} \)
29 \( 1 + 37.4iT - 841T^{2} \)
31 \( 1 + 2.86iT - 961T^{2} \)
37 \( 1 - 33.0iT - 1.36e3T^{2} \)
41 \( 1 - 20.9iT - 1.68e3T^{2} \)
43 \( 1 - 23.9T + 1.84e3T^{2} \)
47 \( 1 - 33.8T + 2.20e3T^{2} \)
53 \( 1 + 37.4iT - 2.80e3T^{2} \)
59 \( 1 + 32.5iT - 3.48e3T^{2} \)
61 \( 1 - 6.58T + 3.72e3T^{2} \)
67 \( 1 + 112. iT - 4.48e3T^{2} \)
71 \( 1 - 110. iT - 5.04e3T^{2} \)
73 \( 1 - 134.T + 5.32e3T^{2} \)
79 \( 1 - 48.8iT - 6.24e3T^{2} \)
83 \( 1 - 44.2T + 6.88e3T^{2} \)
89 \( 1 + 173. iT - 7.92e3T^{2} \)
97 \( 1 - 50.6iT - 9.40e3T^{2} \)
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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

−10.55246960514032327029731949089, −9.650364948972474474944438876439, −8.392434184206949521731570408854, −7.72553579269166777223455474537, −6.58235254538518681412759470110, −5.02284469145782705795797303445, −4.25426610367258153672749014058, −2.59706973989185271710935662266, −1.94322295607821004659603378894, −0.52912365079345627994538898366, 2.13058786689036519503945897068, 4.17416074373615591652601840780, 4.91669766660034349767474299607, 5.64044475585020265486927112425, 6.92581266320034917927749579301, 7.69348852202849080168033150940, 8.449085548993646955290218874676, 9.261249339818676897536785257388, 10.64269380589549778419476300977, 10.91376462287203118821854288705

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