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Properties

Label 2-475-19.18-c2-0-9
Degree $2$
Conductor $475$
Sign $0.370 + 0.928i$
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  + 2.34i·2-s + 5.73i·3-s − 1.51·4-s − 13.4·6-s − 9.56·7-s + 5.83i·8-s − 23.9·9-s + 4.15·11-s − 8.68i·12-s + 4.55i·13-s − 22.4i·14-s − 19.7·16-s + 27.9·17-s − 56.1i·18-s + (7.04 + 17.6i)19-s + ⋯
L(s)  = 1  + 1.17i·2-s + 1.91i·3-s − 0.378·4-s − 2.24·6-s − 1.36·7-s + 0.729i·8-s − 2.65·9-s + 0.377·11-s − 0.723i·12-s + 0.350i·13-s − 1.60i·14-s − 1.23·16-s + 1.64·17-s − 3.11i·18-s + (0.370 + 0.928i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.370 + 0.928i$
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: \((2,\ 475,\ (\ :1),\ 0.370 + 0.928i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.099444799\)
\(L(\frac12)\) \(\approx\) \(1.099444799\)
\(L(2)\) 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 \( 1 + (-7.04 - 17.6i)T \)
good2 \( 1 - 2.34iT - 4T^{2} \)
3 \( 1 - 5.73iT - 9T^{2} \)
7 \( 1 + 9.56T + 49T^{2} \)
11 \( 1 - 4.15T + 121T^{2} \)
13 \( 1 - 4.55iT - 169T^{2} \)
17 \( 1 - 27.9T + 289T^{2} \)
23 \( 1 - 13.9T + 529T^{2} \)
29 \( 1 - 14.6iT - 841T^{2} \)
31 \( 1 + 28.9iT - 961T^{2} \)
37 \( 1 - 30.8iT - 1.36e3T^{2} \)
41 \( 1 + 44.2iT - 1.68e3T^{2} \)
43 \( 1 + 69.2T + 1.84e3T^{2} \)
47 \( 1 + 30.1T + 2.20e3T^{2} \)
53 \( 1 - 54.3iT - 2.80e3T^{2} \)
59 \( 1 + 31.3iT - 3.48e3T^{2} \)
61 \( 1 - 99.9T + 3.72e3T^{2} \)
67 \( 1 + 3.05iT - 4.48e3T^{2} \)
71 \( 1 - 23.0iT - 5.04e3T^{2} \)
73 \( 1 + 21.1T + 5.32e3T^{2} \)
79 \( 1 + 77.7iT - 6.24e3T^{2} \)
83 \( 1 + 93.1T + 6.88e3T^{2} \)
89 \( 1 - 126. iT - 7.92e3T^{2} \)
97 \( 1 - 77.2iT - 9.40e3T^{2} \)
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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

−11.35347072412283913263861052510, −10.21140380862684824373153416523, −9.735049365600413868089973858680, −8.938944132711700304061679569271, −7.995867253428655962527752003922, −6.70488952700975395181783245552, −5.80439954188646801488226010558, −5.14723621357451024162140037326, −3.84145022836352269436104085205, −3.06160665489974597845702390618, 0.45641377103411013123135731274, 1.44242232215690719845716332269, 2.82663354375394769255600925051, 3.32636692966797834012711600333, 5.54522101310656740447355633420, 6.65730120617354490515679299705, 7.04927465274878912912366053874, 8.219087171061088696615553144431, 9.352635436003847128353781222745, 10.15751585793215482840039501040

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