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Properties

Label 1-475-475.86-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.701 + 0.712i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0348 − 0.999i)2-s + (0.719 − 0.694i)3-s + (−0.997 + 0.0697i)4-s + (−0.719 − 0.694i)6-s + (−0.5 − 0.866i)7-s + (0.104 + 0.994i)8-s + (0.0348 − 0.999i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (0.882 − 0.469i)13-s + (−0.848 + 0.529i)14-s + (0.990 − 0.139i)16-s + (0.559 − 0.829i)17-s − 18-s + (−0.961 − 0.275i)21-s + (0.719 − 0.694i)22-s + ⋯
L(s)  = 1  + (−0.0348 − 0.999i)2-s + (0.719 − 0.694i)3-s + (−0.997 + 0.0697i)4-s + (−0.719 − 0.694i)6-s + (−0.5 − 0.866i)7-s + (0.104 + 0.994i)8-s + (0.0348 − 0.999i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (0.882 − 0.469i)13-s + (−0.848 + 0.529i)14-s + (0.990 − 0.139i)16-s + (0.559 − 0.829i)17-s − 18-s + (−0.961 − 0.275i)21-s + (0.719 − 0.694i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.701 + 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6748890716 - 1.610507208i\)
\(L(\frac12)\) \(\approx\) \(-0.6748890716 - 1.610507208i\)
\(L(1)\) \(\approx\) \(0.6497581168 - 0.9519090903i\)
\(L(1)\) \(\approx\) \(0.6497581168 - 0.9519090903i\)

Euler product

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

−24.2393101416362032924115890810, −23.49013924128714858427230656980, −22.136356768913420334229222885996, −21.93661252427173431345518599963, −20.97637538338913639704737978705, −19.634298547304755939112191623273, −18.99708320144725352623258446388, −18.24507927580679197398772344832, −16.8418271552860202634123362654, −16.34054052286937391398274831487, −15.48482247462657880816140116839, −14.84066674937347968828937321014, −13.92933670236382701252100758549, −13.25680096895109827776929981425, −12.03570278754547247493986138538, −10.69738446712426353930577370479, −9.63804922409124778919028732673, −8.82468558083369874594692122819, −8.43587537013917649508907655367, −7.13596317448928071286631504901, −5.99038042698133419963760361659, −5.31975635016984864013506164277, −3.84986246311368956352529404591, −3.38875926425456870140337812466, −1.595185235504564454688319400470, 0.43861757176242721093273603712, 1.370537627518036207124285487501, 2.51879316624099149951696427658, 3.57948578776115634814764871372, 4.24891994371534018333358667190, 5.884330524795705893599832818171, 7.11009174609559969422908179901, 7.943006983603067015310707263184, 9.06394572747842255228080627045, 9.73860696351386067560624451549, 10.716154664713370472686432830305, 11.84016452197817058190576603266, 12.65270332940038160130017937242, 13.37680741637066030509658697564, 14.09796851660024080747911981854, 14.90166448284175765703385048606, 16.34397444076218596960978728947, 17.421339451653004969805624415666, 18.198781480358039135858000062885, 18.96937181289898320079247459439, 19.80178591605449383136859759088, 20.4689370658523661673136847457, 20.8708613534618626887328901742, 22.394469364989883798653419310235, 22.944672170974611538111458364939

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