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Properties

Label 1-475-475.216-r0-0-0
Degree $1$
Conductor $475$
Sign $0.567 - 0.823i$
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  + (−0.978 − 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (−0.104 + 0.994i)6-s + 7-s + (−0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.309 + 0.951i)11-s + (0.309 − 0.951i)12-s + (−0.978 + 0.207i)13-s + (−0.978 − 0.207i)14-s + (0.669 + 0.743i)16-s + (0.913 − 0.406i)17-s + 18-s + (−0.104 − 0.994i)21-s + (−0.104 − 0.994i)22-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (−0.104 + 0.994i)6-s + 7-s + (−0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.309 + 0.951i)11-s + (0.309 − 0.951i)12-s + (−0.978 + 0.207i)13-s + (−0.978 − 0.207i)14-s + (0.669 + 0.743i)16-s + (0.913 − 0.406i)17-s + 18-s + (−0.104 − 0.994i)21-s + (−0.104 − 0.994i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.567 - 0.823i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (216, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.567 - 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8088661694 - 0.4246245413i\)
\(L(\frac12)\) \(\approx\) \(0.8088661694 - 0.4246245413i\)
\(L(1)\) \(\approx\) \(0.7287739972 - 0.2538699487i\)
\(L(1)\) \(\approx\) \(0.7287739972 - 0.2538699487i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.978 - 0.207i)T \)
3 \( 1 + (-0.104 - 0.994i)T \)
7 \( 1 + T \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.978 + 0.207i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.669 - 0.743i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (-0.104 + 0.994i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (-0.104 - 0.994i)T \)
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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

−24.052140171206328452629717321416, −23.30867784724468572449191266490, −21.88789862116133601975189924134, −21.35814544552622842753677941094, −20.541337618572356493047278365994, −19.65588740738058745508337332098, −18.868321286099403833428855133981, −17.67046405966889513239717761577, −17.0939863831642113246814889930, −16.43247757764486582378755697517, −15.39568201876332551307544346349, −14.72459637320594177060111238793, −13.99635552400190772889714808211, −12.1091307838589749244453271473, −11.41928379220954734666524012280, −10.58796065799528536567323246304, −9.86064189341793696406291151489, −8.82100360350200711420299634427, −8.19253465780444266357688229191, −7.117539694371257067724957458494, −5.72149774897309171973808519041, −5.12959799098663825286276264208, −3.67086010173200458139296774615, −2.50887905541506445019765944984, −1.00510256534010873322125448515, 0.95700964863067516857409360231, 1.96007284069215447856014861256, 2.79976584847072294583119403261, 4.58174259379597824742229938003, 5.82857096213045432020118033980, 7.15804611292821819533300158651, 7.45129360078295119556735530959, 8.47763208430396819899115225687, 9.42240766841256399682431176117, 10.50572050536999424152537269614, 11.52134438193979123602672935789, 12.13436532194408431187044049206, 12.88004928145180691760438098472, 14.44549746669874938720798599815, 14.805421051847829902315316105099, 16.37795893335576711820331461580, 17.13593374313570898200291022226, 17.81872824494210578194882929685, 18.43122868481052134534449238661, 19.34168138010144676963677220638, 20.09091998186150279555565194132, 20.831090048878909261693299775338, 21.85414437792169969046242928451, 23.044810897754264351141686420779, 23.89799060144707704592418950621

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