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Properties

Label 1-532-532.531-r1-0-0
Degree $1$
Conductor $532$
Sign $1$
Analytic cond. $57.1713$
Root an. cond. $57.1713$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s − 11-s + 13-s + 15-s − 17-s − 23-s + 25-s − 27-s − 29-s − 31-s + 33-s − 37-s − 39-s + 41-s − 43-s − 45-s + 47-s + 51-s − 53-s + 55-s − 59-s − 61-s − 65-s + 67-s + 69-s + ⋯
L(s)  = 1  − 3-s − 5-s + 9-s − 11-s + 13-s + 15-s − 17-s − 23-s + 25-s − 27-s − 29-s − 31-s + 33-s − 37-s − 39-s + 41-s − 43-s − 45-s + 47-s + 51-s − 53-s + 55-s − 59-s − 61-s − 65-s + 67-s + 69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(57.1713\)
Root analytic conductor: \(57.1713\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{532} (531, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5329433466\)
\(L(\frac12)\) \(\approx\) \(0.5329433466\)
\(L(1)\) \(\approx\) \(0.5448212620\)
\(L(1)\) \(\approx\) \(0.5448212620\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + T \)
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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

−23.32113258135600496062214537827, −22.559199833743978254307764069824, −21.76407935109655769957378106139, −20.71870415341554021949618280082, −19.98279057739425227046842886576, −18.728559723235939669989920060524, −18.33891640554523378800047175101, −17.37415845610870022355542432715, −16.234895203691951701179856933986, −15.83658414952795676599747035782, −15.079096608079677735000927162073, −13.6195329069126924328209406575, −12.78898000881435115831810044308, −11.99643121049878288756355964996, −10.94877065874875245521573365887, −10.73823270513230944956558194719, −9.29967051160146001616696605689, −8.14690410798363726679712054038, −7.34181527697572561766947186476, −6.33349604789714380718764725241, −5.36112802074235802557788879581, −4.36961162971360907940192124601, −3.49962053869465922597475190420, −1.87710644801835542839561842419, −0.399506716779202774373203475210, 0.399506716779202774373203475210, 1.87710644801835542839561842419, 3.49962053869465922597475190420, 4.36961162971360907940192124601, 5.36112802074235802557788879581, 6.33349604789714380718764725241, 7.34181527697572561766947186476, 8.14690410798363726679712054038, 9.29967051160146001616696605689, 10.73823270513230944956558194719, 10.94877065874875245521573365887, 11.99643121049878288756355964996, 12.78898000881435115831810044308, 13.6195329069126924328209406575, 15.079096608079677735000927162073, 15.83658414952795676599747035782, 16.234895203691951701179856933986, 17.37415845610870022355542432715, 18.33891640554523378800047175101, 18.728559723235939669989920060524, 19.98279057739425227046842886576, 20.71870415341554021949618280082, 21.76407935109655769957378106139, 22.559199833743978254307764069824, 23.32113258135600496062214537827

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