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Properties

Label 1-532-532.311-r0-0-0
Degree $1$
Conductor $532$
Sign $0.572 + 0.819i$
Analytic cond. $2.47059$
Root an. cond. $2.47059$
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  + 3-s + (0.5 + 0.866i)5-s + 9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s − 17-s − 23-s + (−0.5 + 0.866i)25-s + 27-s + (−0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + 3-s + (0.5 + 0.866i)5-s + 9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s − 17-s − 23-s + (−0.5 + 0.866i)25-s + 27-s + (−0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.572 + 0.819i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ 0.572 + 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.914010058 + 0.9977863777i\)
\(L(\frac12)\) \(\approx\) \(1.914010058 + 0.9977863777i\)
\(L(1)\) \(\approx\) \(1.552660415 + 0.3938392219i\)
\(L(1)\) \(\approx\) \(1.552660415 + 0.3938392219i\)

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 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−23.616166244187212638301697780537, −22.201657128190592652046577028283, −21.59016587817091488189564288425, −20.680376800201468948562233275604, −20.0325077091280125411264420219, −19.425366087094034076623160701154, −18.21726848860793334160391242505, −17.586744178886155201128379149407, −16.25966539642394767789758304474, −15.88105687324898726647581327363, −14.63308524072704714793585616326, −13.89630852248456627214507410666, −13.119846060675615053932385162019, −12.49260311264231780631611189075, −11.12737599997387350038896400893, −10.13265894361932247230569657651, −9.05845095353274494844297756025, −8.65553066208761440346278247492, −7.719348989576244207889188200578, −6.44007662349350304022216010085, −5.45185569341183083381504993860, −4.26117016476697980648259992112, −3.35333128351311160198963352126, −2.141001890643082638041462639645, −1.08571587321343820702347258740, 1.83673676169111869694516896831, 2.31582124487479001918520691214, 3.69841137145733227548195077813, 4.37726702453869955132556878537, 6.03365721136701099625415836289, 6.89162996949280217924384216204, 7.64468229827437510078159412616, 8.91605038235908660865981360867, 9.53095148705032234449575400522, 10.41321451784590514367154268073, 11.440593298636328332146549098950, 12.55420705372693676038074659543, 13.7395995813507299720534571867, 13.99277898517120483322654032140, 15.11013275559824602541471474373, 15.56582166131662453235558251374, 16.91936962087893561213283338730, 17.87362011731015683970083661402, 18.64126166218656538784905519927, 19.34154798758501452009629410944, 20.32058954673606548721779243372, 20.94481977584059974215902035135, 22.01439554242397058802775388375, 22.46316557042260976791245051320, 23.75065895844098592771450888372

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