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Properties

Label 1-475-475.192-r0-0-0
Degree $1$
Conductor $475$
Sign $0.555 + 0.831i$
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  + (−0.275 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6173860918 + 0.3301563532i\)
\(L(\frac12)\) \(\approx\) \(0.6173860918 + 0.3301563532i\)
\(L(1)\) \(\approx\) \(0.7138616395 + 0.005853402110i\)
\(L(1)\) \(\approx\) \(0.7138616395 + 0.005853402110i\)

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

−23.97077659097838387219349440484, −22.765721362944394025340036877191, −22.58440052411643261481108122857, −21.36326394428217227276731333807, −19.71680127175117180460185443173, −19.14654075874786477861761664391, −18.67366619077449842669284506269, −17.58708355513729529684526653401, −16.77068318960697723153596193201, −16.26319790416986245693726762448, −14.940701337812166494222122756093, −14.21416007419258337785727875387, −13.42961308220857943188093546810, −12.39528144864870078941743017473, −11.72791887327111272693819740975, −10.177161666347423881353699321352, −9.24877175657467856929145404977, −8.440949808388625845434194942667, −7.437156984606185488128928463192, −6.488604116938074213260596668359, −6.07870643180601384513149514941, −4.8678722452558196767546577383, −3.41614432717830529287443774661, −1.95702297423271668952679788833, −0.49840951874095964913065638135, 1.20134814742019377593940404985, 2.95817585147423245955989708990, 3.53237873714361770247086645882, 4.550711909268867408779920499810, 5.54380863468466522491020583752, 6.98799930913848327650888526186, 8.23097456540155519393225235201, 9.335678890612575690995725941247, 9.96246042546468390305463979850, 10.46675070700828936416638347689, 11.71588919139356200304426990324, 12.29427752821679962643546650598, 13.4461337787820166022395831562, 14.37405907673244405748823473075, 15.31867066836545775496866486954, 16.55718073337785000757477069807, 17.03823418287700193456936258837, 17.87766492150834198197843658367, 19.20848828799831745469746073873, 19.851735343261052065191448453062, 20.41229051647339936184411572469, 21.44670954541898361415276863607, 22.14598520421590424726895898606, 22.7889174098193944092903318134, 23.44821387713720498061280740317

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