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Properties

Label 1-475-475.233-r1-0-0
Degree $1$
Conductor $475$
Sign $0.176 + 0.984i$
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  + (0.970 + 0.241i)2-s + (−0.788 − 0.615i)3-s + (0.882 + 0.469i)4-s + (−0.615 − 0.788i)6-s + (0.866 + 0.5i)7-s + (0.743 + 0.669i)8-s + (0.241 + 0.970i)9-s + (0.913 + 0.406i)11-s + (−0.406 − 0.913i)12-s + (−0.275 + 0.961i)13-s + (0.719 + 0.694i)14-s + (0.559 + 0.829i)16-s + (0.529 − 0.848i)17-s + i·18-s + (−0.374 − 0.927i)21-s + (0.788 + 0.615i)22-s + ⋯
L(s)  = 1  + (0.970 + 0.241i)2-s + (−0.788 − 0.615i)3-s + (0.882 + 0.469i)4-s + (−0.615 − 0.788i)6-s + (0.866 + 0.5i)7-s + (0.743 + 0.669i)8-s + (0.241 + 0.970i)9-s + (0.913 + 0.406i)11-s + (−0.406 − 0.913i)12-s + (−0.275 + 0.961i)13-s + (0.719 + 0.694i)14-s + (0.559 + 0.829i)16-s + (0.529 − 0.848i)17-s + i·18-s + (−0.374 − 0.927i)21-s + (0.788 + 0.615i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.176 + 0.984i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.176 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.487970583 + 2.081483577i\)
\(L(\frac12)\) \(\approx\) \(2.487970583 + 2.081483577i\)
\(L(1)\) \(\approx\) \(1.685297207 + 0.4420941387i\)
\(L(1)\) \(\approx\) \(1.685297207 + 0.4420941387i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.970 + 0.241i)T \)
3 \( 1 + (-0.788 - 0.615i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.275 + 0.961i)T \)
17 \( 1 + (0.529 - 0.848i)T \)
23 \( 1 + (-0.898 + 0.438i)T \)
29 \( 1 + (-0.848 + 0.529i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (0.559 + 0.829i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (-0.529 - 0.848i)T \)
53 \( 1 + (0.469 - 0.882i)T \)
59 \( 1 + (0.997 - 0.0697i)T \)
61 \( 1 + (0.438 + 0.898i)T \)
67 \( 1 + (0.927 + 0.374i)T \)
71 \( 1 + (0.990 + 0.139i)T \)
73 \( 1 + (0.275 + 0.961i)T \)
79 \( 1 + (0.615 - 0.788i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (-0.559 + 0.829i)T \)
97 \( 1 + (-0.927 + 0.374i)T \)
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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

−23.288207409670241613698842953634, −22.46379059898816072089838493882, −21.873991760483735088073046292617, −21.03784298032668793529242619079, −20.33597116915011109319303263925, −19.47922069470359130908153773361, −18.15561409484299503166339966853, −17.14446942328215225353319722987, −16.56747220577823139250994942334, −15.47627181121199753956227293769, −14.694419319410906186922503059, −14.067549461525753615807534230623, −12.746370278816734426415211113072, −12.05867328536244055631969869029, −11.09718889609971826906647014810, −10.60072888213032011418813783505, −9.61893347384151716110652536318, −8.09683805014208601336570280865, −6.94732961298476277529573853513, −5.86391079916640401263930447307, −5.250570886131794671860936382935, −4.09090215901174899147969330828, −3.57584845144855157087749684985, −1.83831865762521015086880113702, −0.658384683099595465116934979378, 1.48284236559219117811800644800, 2.19384443176722016038892331267, 3.84011366384236608456249438702, 4.90849638736371676791125173146, 5.54500930458878561807577413015, 6.66945027168786111670349572799, 7.30220111865450526196744721359, 8.34260455411926734970339562093, 9.77159619642946158329187837457, 11.26372923238551745308866449528, 11.682374876536899320443569289173, 12.29155156444329787826766330910, 13.378655135332714837531479000092, 14.28572285070080082099352615087, 14.88910683127267932417580792650, 16.20151480530054093467208803677, 16.76902078708697268081080049425, 17.70563543470872924655220965426, 18.53808064749763731958875421026, 19.632176314651945751415106772443, 20.57958158703706335877986520046, 21.658370750688009354615488604937, 22.10728877297684652399948233783, 23.01604755081939525670600600786, 23.836787375821328016770463493731

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