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Properties

Label 1-475-475.221-r1-0-0
Degree $1$
Conductor $475$
Sign $0.0559 + 0.998i$
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  + (−0.913 + 0.406i)2-s + (0.978 + 0.207i)3-s + (0.669 − 0.743i)4-s + (−0.978 + 0.207i)6-s + 7-s + (−0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)12-s + (−0.913 − 0.406i)13-s + (−0.913 + 0.406i)14-s + (−0.104 − 0.994i)16-s + (0.669 + 0.743i)17-s − 18-s + (0.978 + 0.207i)21-s + (0.978 + 0.207i)22-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.978 + 0.207i)3-s + (0.669 − 0.743i)4-s + (−0.978 + 0.207i)6-s + 7-s + (−0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)12-s + (−0.913 − 0.406i)13-s + (−0.913 + 0.406i)14-s + (−0.104 − 0.994i)16-s + (0.669 + 0.743i)17-s − 18-s + (0.978 + 0.207i)21-s + (0.978 + 0.207i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0559 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0559 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.0559 + 0.998i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.0559 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.373530130 + 1.298682715i\)
\(L(\frac12)\) \(\approx\) \(1.373530130 + 1.298682715i\)
\(L(1)\) \(\approx\) \(1.026931884 + 0.3474968873i\)
\(L(1)\) \(\approx\) \(1.026931884 + 0.3474968873i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.913 + 0.406i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.104 + 0.994i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (-0.669 + 0.743i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.978 - 0.207i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (0.978 + 0.207i)T \)
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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

−23.7980750888802686074005182252, −22.270662644200857641788178666387, −21.25710332275389549822609776250, −20.54736068768086654587902972321, −20.23760375895922788529583905655, −18.85445613587800593301116984775, −18.61763842281954456477301738842, −17.60177932041136575133569645644, −16.74186373838553230318297350738, −15.57206172070089130718528687267, −14.83313082699628888939484287322, −13.91240112463197579809195849826, −12.74202912914615858794199989361, −12.02645618126356100478159808042, −10.95853508018452505433121879844, −9.88571593204171834695623864395, −9.31916917791552690562887154924, −8.05883879850291792013258336351, −7.74348934637695472781805982399, −6.79078967291894449117301012789, −5.0032631124022921427215506948, −3.87733728914837472398501731225, −2.46474515095156617202751663459, −2.05607432252532867572214197511, −0.60539734421776733735695716659, 1.18860067662851793151887654250, 2.20640977903942166210380423235, 3.29185236336236231495790131169, 4.86160293512728727720732120162, 5.70303876211125929154653949621, 7.31928882853994984816644533096, 7.8465053259475980547622664359, 8.54601376542516390690758816852, 9.537479060614003841909193533694, 10.407696804128379160204441392711, 11.1498009592551204504658191386, 12.4783448936917171222917550964, 13.69919812597785425145094103463, 14.66414471317509103681376708029, 15.055406661009117347322868083868, 16.0805832605263117159384653066, 16.938735943537629669891892157484, 17.99175642878441287323758672122, 18.59947414372070664641197094554, 19.63076988807103919802653066812, 20.120147909163034720686944758619, 21.23273397942124065624564010217, 21.624561705174267221937903135969, 23.436950356468568121751774392434, 24.064285015977029821341454834071

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