Properties

Label 1-460-460.187-r0-0-0
Degree $1$
Conductor $460$
Sign $0.229 - 0.973i$
Analytic cond. $2.13623$
Root an. cond. $2.13623$
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  + (0.755 − 0.654i)3-s + (−0.909 − 0.415i)7-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)11-s + (0.909 − 0.415i)13-s + (−0.540 + 0.841i)17-s + (0.841 − 0.540i)19-s + (−0.959 + 0.281i)21-s + (−0.540 − 0.841i)27-s + (−0.841 − 0.540i)29-s + (0.654 − 0.755i)31-s + (0.909 − 0.415i)33-s + (−0.989 − 0.142i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)3-s + (−0.909 − 0.415i)7-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)11-s + (0.909 − 0.415i)13-s + (−0.540 + 0.841i)17-s + (0.841 − 0.540i)19-s + (−0.959 + 0.281i)21-s + (−0.540 − 0.841i)27-s + (−0.841 − 0.540i)29-s + (0.654 − 0.755i)31-s + (0.909 − 0.415i)33-s + (−0.989 − 0.142i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.291610957 - 1.022861328i\)
\(L(\frac12)\) \(\approx\) \(1.291610957 - 1.022861328i\)
\(L(1)\) \(\approx\) \(1.232423502 - 0.4547161346i\)
\(L(1)\) \(\approx\) \(1.232423502 - 0.4547161346i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.755 + 0.654i)T \)
7 \( 1 + (0.909 + 0.415i)T \)
11 \( 1 + (-0.959 - 0.281i)T \)
13 \( 1 + (-0.909 + 0.415i)T \)
17 \( 1 + (0.540 - 0.841i)T \)
19 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (0.989 + 0.142i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (-0.755 + 0.654i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.909 + 0.415i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (0.654 - 0.755i)T \)
67 \( 1 + (-0.281 - 0.959i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.540 + 0.841i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (0.989 + 0.142i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.989 + 0.142i)T \)
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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

−24.42319598564271542835258165186, −22.93816489389584911050375526438, −22.35571735122283009117125695523, −21.57456557176890227624002828642, −20.62558277874480468410468031900, −19.9096543784855335304439489724, −19.07208858804311653911737683302, −18.36491981123460992825161904171, −16.98276731863774453116916657667, −15.98303769581475015085690211900, −15.7646725787107757320240240138, −14.40646423221229398466984672760, −13.86714590457052741795729390365, −12.886382368587706387808923781955, −11.72721504495147586098352478194, −10.81807490389324753312635303461, −9.55005369894240928175182025127, −9.21257933464489044648567138818, −8.23993784653270127703116282673, −6.96251315713512076210920043776, −6.00784386790812460793595614418, −4.758623065821716841453750504835, −3.6138665497432876038944908587, −3.00249405024243125144062542059, −1.56288739129490301845232030571, 0.91570214623819360652065310617, 2.14746917839616029436905252810, 3.44035784129842815748614231360, 4.024968444601669138022036184402, 5.862883811883325939358796035610, 6.689230144746364695141583302008, 7.46958181994110918848606221511, 8.64382096363693312983653597205, 9.32461142881070705953460060255, 10.33223867850248797345691119746, 11.57233537027581551685769887150, 12.54811310981226131944727172222, 13.37537377993893786610106887651, 13.91025764980347838711071080682, 15.10292205224488625256037469846, 15.75348666811437688200496360804, 17.01397329947878549725495618352, 17.74675299499780856593119931833, 18.827187878171846756108073519403, 19.485364108054652033094497814885, 20.1899801488485597973402872893, 20.90273361706091404870085363647, 22.283721336666954065142880969982, 22.815138885180145647225804316584, 23.92952645738468905383298552554

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