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Properties

Label 1-15e2-225.169-r0-0-0
Degree $1$
Conductor $225$
Sign $0.747 + 0.663i$
Analytic cond. $1.04489$
Root an. cond. $1.04489$
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 + (0.669 + 0.743i)4-s + (0.5 + 0.866i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)13-s + (−0.104 − 0.994i)14-s + (−0.104 + 0.994i)16-s + (−0.309 − 0.951i)17-s + (0.309 + 0.951i)19-s + (−0.669 − 0.743i)22-s + (0.104 + 0.994i)23-s + 26-s + (−0.309 + 0.951i)28-s + (−0.978 + 0.207i)29-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (0.5 + 0.866i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)13-s + (−0.104 − 0.994i)14-s + (−0.104 + 0.994i)16-s + (−0.309 − 0.951i)17-s + (0.309 + 0.951i)19-s + (−0.669 − 0.743i)22-s + (0.104 + 0.994i)23-s + 26-s + (−0.309 + 0.951i)28-s + (−0.978 + 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.747 + 0.663i$
Analytic conductor: \(1.04489\)
Root analytic conductor: \(1.04489\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (0:\ ),\ 0.747 + 0.663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7312633951 + 0.2777809056i\)
\(L(\frac12)\) \(\approx\) \(0.7312633951 + 0.2777809056i\)
\(L(1)\) \(\approx\) \(0.7499938357 + 0.06328183410i\)
\(L(1)\) \(\approx\) \(0.7499938357 + 0.06328183410i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.913 - 0.406i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.104 + 0.994i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.978 - 0.207i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (0.978 + 0.207i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (-0.669 + 0.743i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
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

−26.56120994673968983516193491264, −25.5128856444271676761750601911, −24.3225690667986736811264945243, −24.08658254827499835981289191711, −22.72378810154255876285077800905, −21.62353225713466251541349695387, −20.25587909730371836197113030510, −19.80778799443446823182859701368, −18.7778636695409948330649601010, −17.51575919440421648623229238205, −17.12851274238901624262899629447, −16.13133772841696416175624940449, −14.829710274878667424337996037770, −14.31036914487766085402352766656, −12.849665753963018802033747123359, −11.399464602246890253581363840974, −10.72322764975474779043335930412, −9.605110400361660525219580752739, −8.610789989688287880307418509667, −7.53273711502924582760793855206, −6.71467290099821432671287868246, −5.434253780366396320091422842066, −4.053681840778853371149072881260, −2.270861566812223816978503887772, −0.82594710296061483758479193108, 1.54038382141807147237391383682, 2.57530740418991384719278931434, 4.05477497209557110593316384188, 5.575800382918724640893591521089, 7.004780406784377984204009970460, 7.86957472434275466446165276628, 9.24437549501214478778274643648, 9.5668771208032664900233358036, 11.16956393945639027255443193141, 11.83191170320193355132942874395, 12.66383327151717912144918887781, 14.281203146274184252394081638472, 15.22252331520682875332742933732, 16.356656483116821341484001036305, 17.25622442689577182164803022518, 18.14785182331544148191357787340, 18.94733316619393306829700798375, 19.92085538578343673042317311906, 20.76458360494348149520846969402, 21.8017109269125053795717054468, 22.46235909214789844302288416814, 24.12588938102708001259930027881, 24.94400401263701329411824782320, 25.54507415009626789774834654626, 26.87430349676315796033379700536

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