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Properties

Label 1-475-475.104-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: $\chi_{475} (104, \cdot )$
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

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

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