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Properties

Label 1-465-465.224-r1-0-0
Degree $1$
Conductor $465$
Sign $-0.807 - 0.589i$
Analytic cond. $49.9711$
Root an. cond. $49.9711$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.669 − 0.743i)7-s + (0.309 + 0.951i)8-s + (0.978 − 0.207i)11-s + (0.104 − 0.994i)13-s + (0.978 + 0.207i)14-s + (−0.809 − 0.587i)16-s + (−0.978 − 0.207i)17-s + (−0.104 − 0.994i)19-s + (−0.669 + 0.743i)22-s + (0.309 + 0.951i)23-s + (0.5 + 0.866i)26-s + (−0.913 + 0.406i)28-s + (0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.669 − 0.743i)7-s + (0.309 + 0.951i)8-s + (0.978 − 0.207i)11-s + (0.104 − 0.994i)13-s + (0.978 + 0.207i)14-s + (−0.809 − 0.587i)16-s + (−0.978 − 0.207i)17-s + (−0.104 − 0.994i)19-s + (−0.669 + 0.743i)22-s + (0.309 + 0.951i)23-s + (0.5 + 0.866i)26-s + (−0.913 + 0.406i)28-s + (0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.807 - 0.589i$
Analytic conductor: \(49.9711\)
Root analytic conductor: \(49.9711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 465,\ (1:\ ),\ -0.807 - 0.589i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1611593850 - 0.4941295280i\)
\(L(\frac12)\) \(\approx\) \(0.1611593850 - 0.4941295280i\)
\(L(1)\) \(\approx\) \(0.6373965637 - 0.04351016849i\)
\(L(1)\) \(\approx\) \(0.6373965637 - 0.04351016849i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.669 - 0.743i)T \)
11 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (0.104 - 0.994i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.913 - 0.406i)T \)
43 \( 1 + (0.104 + 0.994i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.978 - 0.207i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−24.275063391996909930418005186682, −22.88665084501305775964114746775, −22.069859724385366572672290861235, −21.50739482420322693296973059665, −20.40797844878409515515243521979, −19.666450213023679060983964873613, −18.856950894577641401326423557088, −18.29356444283858680642487081070, −17.11658723101055338288150685531, −16.50953282588268048109763717813, −15.60845865263448153707711365691, −14.51441475401951185006765175469, −13.31939012010979582018613048624, −12.308029002116256623900399659091, −11.80866837737639423477301034582, −10.73360088671716037283879146334, −9.72291894424097983739306077494, −8.98731570682880320415019965446, −8.31573353706867106682834695076, −6.79617467673364159382997445825, −6.3643323343600213845692345740, −4.54994032813225607383686670315, −3.54308298449007201432722475566, −2.39200704271145029784521745963, −1.407208273860028475317715453630, 0.19979419928242942029260944549, 1.156195695663866191227426975883, 2.70650059516113851016222051160, 4.04912783471355132699125113680, 5.31156892894791535604841106158, 6.48022447664305400675217762997, 7.01624067461434939130424192295, 8.125170210631779762358336767609, 9.11098452922787935171202239004, 9.84576684779757733673472068735, 10.818199708571521865169761057373, 11.59603127357271918803651463688, 13.1081052469811192444328855050, 13.78579935352661216505583353504, 14.92109812575013558541357064296, 15.69531772160139900188013902934, 16.48320485868794434375410678149, 17.46099417787656229062324483225, 17.82920773614027747984763670456, 19.236139471255202790035596110777, 19.72233829115107750339080476754, 20.3414760968180591502437226833, 21.74572753077127205237001089458, 22.755938205316159317122273738422, 23.36782142002062400083827317156

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