↑

Properties

Label 1-475-475.188-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.243 - 0.969i$
Analytic cond. $51.0458$
Root an. cond. $51.0458$
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  + (−0.999 − 0.0348i)2-s + (0.694 − 0.719i)3-s + (0.997 + 0.0697i)4-s + (−0.719 + 0.694i)6-s + (0.866 + 0.5i)7-s + (−0.994 − 0.104i)8-s + (−0.0348 − 0.999i)9-s + (0.669 − 0.743i)11-s + (0.743 − 0.669i)12-s + (0.469 − 0.882i)13-s + (−0.848 − 0.529i)14-s + (0.990 + 0.139i)16-s + (0.829 − 0.559i)17-s + i·18-s + (0.961 − 0.275i)21-s + (−0.694 + 0.719i)22-s + ⋯
L(s)  = 1  + (−0.999 − 0.0348i)2-s + (0.694 − 0.719i)3-s + (0.997 + 0.0697i)4-s + (−0.719 + 0.694i)6-s + (0.866 + 0.5i)7-s + (−0.994 − 0.104i)8-s + (−0.0348 − 0.999i)9-s + (0.669 − 0.743i)11-s + (0.743 − 0.669i)12-s + (0.469 − 0.882i)13-s + (−0.848 − 0.529i)14-s + (0.990 + 0.139i)16-s + (0.829 − 0.559i)17-s + i·18-s + (0.961 − 0.275i)21-s + (−0.694 + 0.719i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.243 - 0.969i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.243 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.196367430 - 1.534192448i\)
\(L(\frac12)\) \(\approx\) \(1.196367430 - 1.534192448i\)
\(L(1)\) \(\approx\) \(0.9809575957 - 0.4337932743i\)
\(L(1)\) \(\approx\) \(0.9809575957 - 0.4337932743i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.999 - 0.0348i)T \)
3 \( 1 + (0.694 - 0.719i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (0.469 - 0.882i)T \)
17 \( 1 + (0.829 - 0.559i)T \)
23 \( 1 + (-0.927 - 0.374i)T \)
29 \( 1 + (-0.559 + 0.829i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.990 + 0.139i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (-0.829 - 0.559i)T \)
53 \( 1 + (0.0697 - 0.997i)T \)
59 \( 1 + (0.615 - 0.788i)T \)
61 \( 1 + (-0.374 + 0.927i)T \)
67 \( 1 + (0.275 - 0.961i)T \)
71 \( 1 + (-0.241 + 0.970i)T \)
73 \( 1 + (-0.469 - 0.882i)T \)
79 \( 1 + (0.719 + 0.694i)T \)
83 \( 1 + (0.406 + 0.913i)T \)
89 \( 1 + (-0.990 + 0.139i)T \)
97 \( 1 + (-0.275 - 0.961i)T \)
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

−24.1665786095204627489420996220, −23.10551309514774872098122991707, −21.78593643443046548470812598225, −20.93647194379535275108339755773, −20.53610826354419232438876445385, −19.53056057994522599202404554520, −18.9476511340057216776171937272, −17.70230125252176324303352227147, −17.060070577331780504452986961466, −16.20807618839078427097433774125, −15.30791063717633702629707690206, −14.49149749190589142129067845309, −13.82401689674951093834986013410, −12.17230236776202135945584005891, −11.32413548928200916207877258886, −10.3892472781639568488742085860, −9.686320808063978304074708042490, −8.79494285024554527696995060117, −7.968774925461635055420744761795, −7.20199040953207472940657365570, −5.90200836392773138006573514744, −4.470592587441737478419407407159, −3.62398384784154076477008131554, −2.133532706045111786464921786504, −1.35031627275154867284499104994, 0.67414245283701891765436385875, 1.5569302907348987757906351163, 2.657493899704184821155901154811, 3.61945144642786988861222605776, 5.55589269061565882237857491620, 6.43005551651105118252279987033, 7.60874228643145867410044779246, 8.23675519437067233888489439976, 8.89292642921914936689076165721, 9.890610958525644388333924400525, 11.10987635913336940635217795817, 11.86548790938985678325161228638, 12.69341204894028819349557327386, 14.03505612458252344998224280721, 14.68073706565226487159936829328, 15.64698071625013582961974908674, 16.62467058283217718839063476596, 17.8000392655202583516944576296, 18.16404301545079329656435542671, 19.05310011859553353170753887139, 19.74821909864696710440486741931, 20.726495893344118603173710665833, 21.15371039667784441903552280133, 22.50798082705307197079491012544, 23.83247873571238246888921613207

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