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Properties

Label 1-475-475.259-r1-0-0
Degree $1$
Conductor $475$
Sign $0.194 - 0.980i$
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  + (−0.978 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.194 - 0.980i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.194 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3820010853 - 0.3138284518i\)
\(L(\frac12)\) \(\approx\) \(0.3820010853 - 0.3138284518i\)
\(L(1)\) \(\approx\) \(0.4735442043 - 0.1458624128i\)
\(L(1)\) \(\approx\) \(0.4735442043 - 0.1458624128i\)

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

−24.07835512687037506534182675580, −22.682153708155617181596414666998, −22.14962507410901428910669050326, −21.1750984369231351521081684919, −20.09759416650986040394477151786, −19.67316028919045476000703424644, −18.696965665061143426393121770731, −17.66288233311316913298597692921, −16.60180665190694996266952113234, −16.41220674440221606330908375655, −15.37473526808880520774766242807, −14.68768602289094684547592322185, −13.486069754747709275902114397836, −12.07122499856194657770638976212, −11.25979435153346446721273457091, −10.3261562125051450262354986018, −9.61736892698707098983195583427, −8.94103999007525849882767863819, −7.95264232899176707542132211457, −6.605482124716956036665104424164, −5.964217014671847982425439511362, −4.71876448930742158517571731970, −3.32894094394584124534704774056, −2.48001094743417698356268990534, −0.524145528698469033018336350980, 0.34034939759800556360627918375, 1.82449406373313434041491087867, 2.501475945911874082184388564609, 3.857299559354019955231785809999, 5.645360903949123522769351074114, 6.79561430982166705145444028106, 7.12899647756493245401307394499, 8.2395826730277335623797528857, 9.299163549247951065253543419756, 9.97584137586946285266849906127, 11.15100658171321778325492699129, 12.168335829914080255818206776421, 12.593344274606410773177892852047, 13.65091999658963790957205090352, 14.946049598945863640510705462976, 15.83873256239165455774567689293, 16.994630690671556896928815781351, 17.436224151130550925560394390219, 18.310436996119960599937444093508, 19.290448714031744869616694061883, 19.68769793783703640662162478171, 20.39112000043520761662604902989, 21.807881281655393613295902512, 22.539940663751898798606797071745, 23.582504664290405870776093947933

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