↑

Properties

Label 1-575-575.279-r0-0-0
Degree $1$
Conductor $575$
Sign $-0.833 + 0.552i$
Analytic cond. $2.67028$
Root an. cond. $2.67028$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.941 − 0.336i)2-s + (−0.516 + 0.856i)3-s + (0.774 + 0.633i)4-s + (0.774 − 0.633i)6-s + (−0.415 + 0.909i)7-s + (−0.516 − 0.856i)8-s + (−0.466 − 0.884i)9-s + (0.610 + 0.791i)11-s + (−0.941 + 0.336i)12-s + (0.870 + 0.491i)13-s + (0.696 − 0.717i)14-s + (0.198 + 0.980i)16-s + (−0.774 + 0.633i)17-s + (0.142 + 0.989i)18-s + (−0.254 − 0.967i)19-s + ⋯
L(s)  = 1  + (−0.941 − 0.336i)2-s + (−0.516 + 0.856i)3-s + (0.774 + 0.633i)4-s + (0.774 − 0.633i)6-s + (−0.415 + 0.909i)7-s + (−0.516 − 0.856i)8-s + (−0.466 − 0.884i)9-s + (0.610 + 0.791i)11-s + (−0.941 + 0.336i)12-s + (0.870 + 0.491i)13-s + (0.696 − 0.717i)14-s + (0.198 + 0.980i)16-s + (−0.774 + 0.633i)17-s + (0.142 + 0.989i)18-s + (−0.254 − 0.967i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $-0.833 + 0.552i$
Analytic conductor: \(2.67028\)
Root analytic conductor: \(2.67028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{575} (279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 575,\ (0:\ ),\ -0.833 + 0.552i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1539735554 + 0.5111889141i\)
\(L(\frac12)\) \(\approx\) \(0.1539735554 + 0.5111889141i\)
\(L(1)\) \(\approx\) \(0.5092100544 + 0.2411850286i\)
\(L(1)\) \(\approx\) \(0.5092100544 + 0.2411850286i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.941 - 0.336i)T \)
3 \( 1 + (-0.516 + 0.856i)T \)
7 \( 1 + (-0.415 + 0.909i)T \)
11 \( 1 + (0.610 + 0.791i)T \)
13 \( 1 + (0.870 + 0.491i)T \)
17 \( 1 + (-0.774 + 0.633i)T \)
19 \( 1 + (-0.254 - 0.967i)T \)
29 \( 1 + (-0.254 + 0.967i)T \)
31 \( 1 + (0.516 + 0.856i)T \)
37 \( 1 + (0.466 + 0.884i)T \)
41 \( 1 + (0.696 + 0.717i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.993 - 0.113i)T \)
59 \( 1 + (-0.870 - 0.491i)T \)
61 \( 1 + (0.974 - 0.226i)T \)
67 \( 1 + (0.0285 - 0.999i)T \)
71 \( 1 + (-0.564 - 0.825i)T \)
73 \( 1 + (0.998 + 0.0570i)T \)
79 \( 1 + (-0.736 + 0.676i)T \)
83 \( 1 + (-0.897 - 0.441i)T \)
89 \( 1 + (0.0855 + 0.996i)T \)
97 \( 1 + (-0.897 + 0.441i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.01156397766133254985534585845, −22.463809137495687759380436360317, −20.89547515681833763872985521645, −20.15583908961636966186322313628, −19.22126949693367105122582135263, −18.7937399759098372017131360584, −17.74742343695607696173897537541, −17.17382717892660496202250838374, −16.38120812623102671126689184255, −15.73904068320037301707623479547, −14.288035740660978192025399698075, −13.6378604100496625164267320078, −12.640388085061794745572518343852, −11.39504829827172775135601664752, −10.98455696839374837255028539232, −9.97766383551702781534375334772, −8.85850782535100974456773764699, −7.94597396863627699198722909481, −7.2105847618589863386365665134, −6.21562370149611552005372135421, −5.80668477069376993800961353786, −4.0908888399371703935333650678, −2.64821459313549788470311073676, −1.33919227639647038141768108319, −0.444585429431142802543081375243, 1.44964300886948196736501054828, 2.72480709007864697307248107490, 3.78207799141734528056676702773, 4.839149905901672630303267784320, 6.32810171334735669087175282538, 6.66341435159448595093069774193, 8.34800493800812594029635873308, 9.125704889078994207200196477079, 9.59479026641560422047157984292, 10.735685732685749859489781444889, 11.357662039370929828533396857161, 12.1970064019952283767615148318, 13.01592075480659079050850255665, 14.68724512601474743627950898727, 15.49576321716412607276780921581, 16.0373105009573614545674768898, 16.960596515174230531069985445543, 17.74259944000038690076639767952, 18.3798260723201548317485799806, 19.5202756160581619113489783433, 20.11444102119204966001835172671, 21.16976511585181445260914801094, 21.75502945642780108873040500041, 22.41041614882523288230498370582, 23.491999741912100108163213700923

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