Properties

Label 2-475-95.94-c2-0-56
Degree $2$
Conductor $475$
Sign $-0.895 + 0.445i$
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  + 2.92·2-s − 2.02·3-s + 4.57·4-s − 5.92·6-s − 8.32i·7-s + 1.68·8-s − 4.90·9-s − 17.3·11-s − 9.25·12-s − 5.27·13-s − 24.3i·14-s − 13.3·16-s + 19.0i·17-s − 14.3·18-s + (11.4 − 15.1i)19-s + ⋯
L(s)  = 1  + 1.46·2-s − 0.674·3-s + 1.14·4-s − 0.987·6-s − 1.18i·7-s + 0.211·8-s − 0.545·9-s − 1.57·11-s − 0.771·12-s − 0.405·13-s − 1.74i·14-s − 0.834·16-s + 1.12i·17-s − 0.798·18-s + (0.601 − 0.798i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.895 + 0.445i$
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: \((2,\ 475,\ (\ :1),\ -0.895 + 0.445i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9522396143\)
\(L(\frac12)\) \(\approx\) \(0.9522396143\)
\(L(2)\) 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 \( 1 - 2.92T + 4T^{2} \)
3 \( 1 + 2.02T + 9T^{2} \)
7 \( 1 + 8.32iT - 49T^{2} \)
11 \( 1 + 17.3T + 121T^{2} \)
13 \( 1 + 5.27T + 169T^{2} \)
17 \( 1 - 19.0iT - 289T^{2} \)
23 \( 1 + 3.32iT - 529T^{2} \)
29 \( 1 + 37.4iT - 841T^{2} \)
31 \( 1 - 2.86iT - 961T^{2} \)
37 \( 1 + 33.0T + 1.36e3T^{2} \)
41 \( 1 + 20.9iT - 1.68e3T^{2} \)
43 \( 1 - 23.9iT - 1.84e3T^{2} \)
47 \( 1 + 33.8iT - 2.20e3T^{2} \)
53 \( 1 + 37.4T + 2.80e3T^{2} \)
59 \( 1 + 32.5iT - 3.48e3T^{2} \)
61 \( 1 - 6.58T + 3.72e3T^{2} \)
67 \( 1 - 112.T + 4.48e3T^{2} \)
71 \( 1 + 110. iT - 5.04e3T^{2} \)
73 \( 1 - 134. iT - 5.32e3T^{2} \)
79 \( 1 - 48.8iT - 6.24e3T^{2} \)
83 \( 1 - 44.2iT - 6.88e3T^{2} \)
89 \( 1 + 173. iT - 7.92e3T^{2} \)
97 \( 1 + 50.6T + 9.40e3T^{2} \)
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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

−10.78391988543806691735736511642, −9.933722746513353594972495597219, −8.369147579237597942820168330632, −7.33200512470299100638036352205, −6.37175439788343431122019869386, −5.42094795194750473308990227976, −4.78648853935656548281946795489, −3.68726874125634701377765352510, −2.53517112711080425485725528647, −0.23506160868411449556061648286, 2.47106220586556873036161719870, 3.20996108225112757316169932119, 5.00442685251702936809253819687, 5.27235320784406179000097946354, 6.00718962806656820859278081747, 7.18925422209640986136413831990, 8.398778706870864697315025952640, 9.470529762127697630742346975927, 10.69475333764912086182915248002, 11.52451611264985153146035792018

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