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Properties

Label 1-4563-4563.383-r0-0-0
Degree $1$
Conductor $4563$
Sign $0.865 + 0.501i$
Analytic cond. $21.1904$
Root an. cond. $21.1904$
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  + (0.891 − 0.452i)2-s + (0.589 − 0.807i)4-s + (0.213 + 0.977i)5-s + (−0.186 − 0.982i)7-s + (0.160 − 0.987i)8-s + (0.632 + 0.774i)10-s + (−0.291 + 0.956i)11-s + (−0.611 − 0.791i)14-s + (−0.303 − 0.952i)16-s + (−0.354 − 0.935i)17-s + (−0.866 + 0.5i)19-s + (0.914 + 0.404i)20-s + (0.173 + 0.984i)22-s + (0.173 + 0.984i)23-s + (−0.909 + 0.416i)25-s + ⋯
L(s)  = 1  + (0.891 − 0.452i)2-s + (0.589 − 0.807i)4-s + (0.213 + 0.977i)5-s + (−0.186 − 0.982i)7-s + (0.160 − 0.987i)8-s + (0.632 + 0.774i)10-s + (−0.291 + 0.956i)11-s + (−0.611 − 0.791i)14-s + (−0.303 − 0.952i)16-s + (−0.354 − 0.935i)17-s + (−0.866 + 0.5i)19-s + (0.914 + 0.404i)20-s + (0.173 + 0.984i)22-s + (0.173 + 0.984i)23-s + (−0.909 + 0.416i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $0.865 + 0.501i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ 0.865 + 0.501i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.485163011 + 0.6677316229i\)
\(L(\frac12)\) \(\approx\) \(2.485163011 + 0.6677316229i\)
\(L(1)\) \(\approx\) \(1.647833235 - 0.1897198532i\)
\(L(1)\) \(\approx\) \(1.647833235 - 0.1897198532i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.891 - 0.452i)T \)
5 \( 1 + (0.213 + 0.977i)T \)
7 \( 1 + (-0.186 - 0.982i)T \)
11 \( 1 + (-0.291 + 0.956i)T \)
17 \( 1 + (-0.354 - 0.935i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.452 + 0.891i)T \)
31 \( 1 + (0.579 + 0.815i)T \)
37 \( 1 + (-0.903 + 0.428i)T \)
41 \( 1 + (0.0268 - 0.999i)T \)
43 \( 1 + (0.815 + 0.579i)T \)
47 \( 1 + (-0.107 - 0.994i)T \)
53 \( 1 + (0.354 + 0.935i)T \)
59 \( 1 + (0.952 + 0.303i)T \)
61 \( 1 + (0.859 + 0.511i)T \)
67 \( 1 + (0.440 + 0.897i)T \)
71 \( 1 + (0.663 - 0.748i)T \)
73 \( 1 + (-0.600 + 0.799i)T \)
79 \( 1 + (-0.404 + 0.914i)T \)
83 \( 1 + (-0.852 + 0.523i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.739 + 0.673i)T \)
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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

−17.92223268671363619021464579727, −17.27596396623930192668429845661, −16.74231810197491151631717707976, −15.94202559723478841706159689290, −15.60806530978154059815095424554, −14.84913118675691221135047796743, −14.12413888428456494074646828777, −13.27297436026576155496848179653, −12.891742651097192756726339569903, −12.352732439014945757622931919369, −11.56454627535489468999482167396, −10.951968193843422848133997148426, −9.959999251203474183810015124956, −8.92815433267574806752872382880, −8.4146704779748294464189075531, −8.09096572929365697573011120921, −6.7999720263346248912702204157, −6.03742397029805879960631292679, −5.79104703711315336624718727763, −4.8184779937764345319891889958, −4.32511169893963962065493251758, −3.39197390828210555879623283269, −2.450999380958234638488017060835, −1.96350585463457798337774540554, −0.498187237484424094804145842536, 1.052685655525703835109111352614, 1.98273867796986588923138669963, 2.67186394282912401979866069211, 3.46259787504243290213339978400, 4.09092873406723980282281576372, 4.88798744644223791968947368687, 5.61252284864996976804737077054, 6.61716562074506519678095418808, 7.04146672778223121507851644727, 7.49127599881120609624935097611, 8.80573892791205302259347322644, 9.93155963502092188688953979707, 10.15513970146675337931805717305, 10.809946127134931323787248801331, 11.4894088851005381311382098132, 12.240734921999707611753930620703, 12.98713078522211865655076874727, 13.62076581840199302111297757407, 14.20031843081207459404954450699, 14.67232346704259747316684105288, 15.576137475484456736825292042198, 15.901525298930359697835961384092, 17.01898240487931684663610648372, 17.665367139614701624534755538471, 18.35490155324622419708549978543

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