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Properties

Label 1-475-475.169-r0-0-0
Degree $1$
Conductor $475$
Sign $0.998 - 0.0576i$
Analytic cond. $2.20589$
Root an. cond. $2.20589$
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  + (−0.961 + 0.275i)2-s + (−0.990 − 0.139i)3-s + (0.848 − 0.529i)4-s + (0.990 − 0.139i)6-s + (0.5 − 0.866i)7-s + (−0.669 + 0.743i)8-s + (0.961 + 0.275i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (0.719 − 0.694i)13-s + (−0.241 + 0.970i)14-s + (0.438 − 0.898i)16-s + (−0.0348 + 0.999i)17-s − 18-s + (−0.615 + 0.788i)21-s + (−0.990 − 0.139i)22-s + ⋯
L(s)  = 1  + (−0.961 + 0.275i)2-s + (−0.990 − 0.139i)3-s + (0.848 − 0.529i)4-s + (0.990 − 0.139i)6-s + (0.5 − 0.866i)7-s + (−0.669 + 0.743i)8-s + (0.961 + 0.275i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (0.719 − 0.694i)13-s + (−0.241 + 0.970i)14-s + (0.438 − 0.898i)16-s + (−0.0348 + 0.999i)17-s − 18-s + (−0.615 + 0.788i)21-s + (−0.990 − 0.139i)22-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.998 - 0.0576i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.998 - 0.0576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7427604907 + 0.02143218106i\)
\(L(\frac12)\) \(\approx\) \(0.7427604907 + 0.02143218106i\)
\(L(1)\) \(\approx\) \(0.6443648381 + 0.0009010257192i\)
\(L(1)\) \(\approx\) \(0.6443648381 + 0.0009010257192i\)

Euler product

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

−23.995908269289676221330625520238, −22.885288937571635084628275374135, −21.88761961668445965181337329835, −21.32458899754202588991359585478, −20.53032313943977894659040811634, −19.2384487757341733826074323818, −18.568543881960527088259857975806, −17.935062660373239045649137303071, −17.03480994889140842806725582398, −16.31936170723220474539531535110, −15.597827865706968042010553232438, −14.52279002359109098816789111142, −13.056850530508690653599929946848, −11.99241441038249622513602952709, −11.410204477595426940958851109386, −10.917595230492438115485494903323, −9.461067889888973348818072007723, −9.07898778819997063268644077581, −7.82852881599773254071005413731, −6.68415934836726304443770774251, −6.010807710648157232623045012237, −4.76101208098828762498178256055, −3.48048611343619379246826705342, −2.01988106264991818428974682287, −0.943763559106604369161772099417, 0.966094488637357334972891309511, 1.692461636967465129062061760093, 3.653641396089025251769694955263, 4.921538308965751698598646752983, 5.997534720986561068370880864681, 6.8395082200649754345790915139, 7.597643673913129696870206436455, 8.65439599543511702518601804669, 9.80411591107055095550705340436, 10.77169515176433427248287374501, 11.10916912652189340143396221835, 12.240314239684986270379375491771, 13.2509881425050685099133735873, 14.60462490100805194963663724512, 15.34238613226726789290474026202, 16.5543463554958957151257917228, 16.96244544029328297004847560993, 17.75331149377962122274296578931, 18.36732763055868964060792476108, 19.475021743848957846201567166759, 20.2262482292376242885154288371, 21.14008959168522222066654814474, 22.22852767585495753697777255793, 23.31871862483437661124395051245, 23.720941034060514811070313015085

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