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Properties

Label 1-475-475.17-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.0576 - 0.998i$
Analytic cond. $51.0458$
Root an. cond. $51.0458$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.275 + 0.961i)2-s + (0.139 − 0.990i)3-s + (−0.848 + 0.529i)4-s + (0.990 − 0.139i)6-s + (−0.866 − 0.5i)7-s + (−0.743 − 0.669i)8-s + (−0.961 − 0.275i)9-s + (0.913 + 0.406i)11-s + (0.406 + 0.913i)12-s + (0.694 + 0.719i)13-s + (0.241 − 0.970i)14-s + (0.438 − 0.898i)16-s + (0.999 + 0.0348i)17-s − i·18-s + (−0.615 + 0.788i)21-s + (−0.139 + 0.990i)22-s + ⋯
L(s)  = 1  + (0.275 + 0.961i)2-s + (0.139 − 0.990i)3-s + (−0.848 + 0.529i)4-s + (0.990 − 0.139i)6-s + (−0.866 − 0.5i)7-s + (−0.743 − 0.669i)8-s + (−0.961 − 0.275i)9-s + (0.913 + 0.406i)11-s + (0.406 + 0.913i)12-s + (0.694 + 0.719i)13-s + (0.241 − 0.970i)14-s + (0.438 − 0.898i)16-s + (0.999 + 0.0348i)17-s − i·18-s + (−0.615 + 0.788i)21-s + (−0.139 + 0.990i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0576 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0576 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.0576 - 0.998i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.0576 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7168408117 - 0.7594385079i\)
\(L(\frac12)\) \(\approx\) \(0.7168408117 - 0.7594385079i\)
\(L(1)\) \(\approx\) \(0.9789146019 + 0.06860442540i\)
\(L(1)\) \(\approx\) \(0.9789146019 + 0.06860442540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.275 + 0.961i)T \)
3 \( 1 + (0.139 - 0.990i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (0.694 + 0.719i)T \)
17 \( 1 + (0.999 + 0.0348i)T \)
23 \( 1 + (-0.0697 + 0.997i)T \)
29 \( 1 + (-0.0348 - 0.999i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.438 - 0.898i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (-0.999 + 0.0348i)T \)
53 \( 1 + (-0.529 - 0.848i)T \)
59 \( 1 + (-0.559 - 0.829i)T \)
61 \( 1 + (-0.997 - 0.0697i)T \)
67 \( 1 + (0.788 - 0.615i)T \)
71 \( 1 + (-0.374 - 0.927i)T \)
73 \( 1 + (-0.694 + 0.719i)T \)
79 \( 1 + (-0.990 - 0.139i)T \)
83 \( 1 + (-0.207 + 0.978i)T \)
89 \( 1 + (-0.438 - 0.898i)T \)
97 \( 1 + (-0.788 - 0.615i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.31586178295224486090303669847, −22.73622772639272504688931380283, −21.9394912494715312327313242112, −21.46094414319758294083875210010, −20.34374022193129666642834943022, −19.89445343732441835825884906073, −18.90589623862376478165285287712, −18.118642508601900680707638892672, −16.77493278321827466509178264847, −16.13235431661219499166750960637, −14.92795202678502326359484154776, −14.40464487710924129799332439248, −13.282670197005419049908717740236, −12.40559050404238585733771737840, −11.470482225459687356581173154444, −10.6071692711808160227085786420, −9.79704872629336878699721476485, −9.05431525291901519428168064833, −8.27558949459594971331033971301, −6.27033862108416646226338579173, −5.55466207209516588501967680513, −4.41383243773568553075498648266, −3.34022647339807711857841063647, −2.94221377425521042973441738851, −1.21568674493254775294537189911, 0.26515917131305650446933577725, 1.59565563414519714516783493318, 3.319274208636548401665458457260, 4.03127624170407569069752715223, 5.62777261461547684961817020283, 6.38583524548062009157039740404, 7.12526219145857920398724574916, 7.8724106373956326878205935862, 9.05272866805644444208814650731, 9.702896565078965970517819039779, 11.41752720085514412170205358535, 12.34755868394405235971768619494, 13.10866168021170956210877859252, 13.9156172234882940239772321672, 14.49500836045866844980088781115, 15.65981663289051022378690737553, 16.67916443290073464928063364935, 17.17151968252378220962781469066, 18.196947450320256717338366442518, 19.023273851138954040831714763270, 19.71543818733762015068454400933, 20.889185142854672105478490180280, 22.027473669849357953217887749771, 22.99974172546552862312724876825, 23.34322317815464594913543473571

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