Properties

Label 2-475-95.94-c2-0-57
Degree $2$
Conductor $475$
Sign $-0.999 + 0.0109i$
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  + 1.88·2-s + 0.840·3-s − 0.461·4-s + 1.58·6-s − 2.31i·7-s − 8.39·8-s − 8.29·9-s − 8.28·11-s − 0.387·12-s − 7.20·13-s − 4.35i·14-s − 13.9·16-s + 6.37i·17-s − 15.6·18-s + (−8.31 + 17.0i)19-s + ⋯
L(s)  = 1  + 0.940·2-s + 0.280·3-s − 0.115·4-s + 0.263·6-s − 0.330i·7-s − 1.04·8-s − 0.921·9-s − 0.753·11-s − 0.0323·12-s − 0.554·13-s − 0.310i·14-s − 0.871·16-s + 0.375i·17-s − 0.866·18-s + (−0.437 + 0.899i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0109i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.999 + 0.0109i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (474, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ -0.999 + 0.0109i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.07924567033\)
\(L(\frac12)\) \(\approx\) \(0.07924567033\)
\(L(2)\) 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 \( 1 + (8.31 - 17.0i)T \)
good2 \( 1 - 1.88T + 4T^{2} \)
3 \( 1 - 0.840T + 9T^{2} \)
7 \( 1 + 2.31iT - 49T^{2} \)
11 \( 1 + 8.28T + 121T^{2} \)
13 \( 1 + 7.20T + 169T^{2} \)
17 \( 1 - 6.37iT - 289T^{2} \)
23 \( 1 + 14.5iT - 529T^{2} \)
29 \( 1 - 16.9iT - 841T^{2} \)
31 \( 1 + 39.5iT - 961T^{2} \)
37 \( 1 + 8.68T + 1.36e3T^{2} \)
41 \( 1 + 41.6iT - 1.68e3T^{2} \)
43 \( 1 + 3.51iT - 1.84e3T^{2} \)
47 \( 1 - 68.8iT - 2.20e3T^{2} \)
53 \( 1 + 89.1T + 2.80e3T^{2} \)
59 \( 1 + 95.2iT - 3.48e3T^{2} \)
61 \( 1 + 12.8T + 3.72e3T^{2} \)
67 \( 1 + 65.8T + 4.48e3T^{2} \)
71 \( 1 - 2.56iT - 5.04e3T^{2} \)
73 \( 1 - 98.0iT - 5.32e3T^{2} \)
79 \( 1 - 27.1iT - 6.24e3T^{2} \)
83 \( 1 - 12.4iT - 6.88e3T^{2} \)
89 \( 1 + 25.1iT - 7.92e3T^{2} \)
97 \( 1 - 62.6T + 9.40e3T^{2} \)
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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

−10.44496954524756730805263080931, −9.425035616508379368532755162224, −8.491507168308555886160862597455, −7.68518550661797545287648336918, −6.28771297918216709605663697282, −5.50495795510728647028497230366, −4.50488421204362971041207936057, −3.46934608963509693717371748474, −2.41068916210648224586579698208, −0.02104356100389574521055850350, 2.50720222474290641257190919329, 3.28881955248511878824803099632, 4.69409302275567089872974913450, 5.36502873451097110217938513630, 6.33163910928635651595514939868, 7.60101107246241324079222078196, 8.676104550570739078306816908067, 9.267453368654441375838254975026, 10.42380141848100162614855102496, 11.55617266469702401044613169989

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