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Properties

Label 2-2151-2151.238-c0-0-3
Degree $2$
Conductor $2151$
Sign $0.438 - 0.898i$
Analytic cond. $1.07348$
Root an. cond. $1.03609$
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  + (−0.990 − 1.71i)2-s + (0.669 + 0.743i)3-s + (−1.46 + 2.53i)4-s + (−0.848 + 1.46i)5-s + (0.612 − 1.88i)6-s + 3.80·8-s + (−0.104 + 0.994i)9-s + 3.35·10-s + (0.997 + 1.72i)11-s + (−2.85 + 0.607i)12-s + (−1.65 + 0.352i)15-s + (−2.30 − 3.99i)16-s + 0.347·17-s + (1.80 − 0.805i)18-s + (−2.47 − 4.29i)20-s + ⋯
L(s)  = 1  + (−0.990 − 1.71i)2-s + (0.669 + 0.743i)3-s + (−1.46 + 2.53i)4-s + (−0.848 + 1.46i)5-s + (0.612 − 1.88i)6-s + 3.80·8-s + (−0.104 + 0.994i)9-s + 3.35·10-s + (0.997 + 1.72i)11-s + (−2.85 + 0.607i)12-s + (−1.65 + 0.352i)15-s + (−2.30 − 3.99i)16-s + 0.347·17-s + (1.80 − 0.805i)18-s + (−2.47 − 4.29i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $0.438 - 0.898i$
Analytic conductor: \(1.07348\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :0),\ 0.438 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6788824592\)
\(L(\frac12)\) \(\approx\) \(0.6788824592\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 - 0.743i)T \)
239 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.990 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.848 - 1.46i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.997 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - 0.347T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.241 - 0.419i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.615 + 1.06i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.61T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.719 - 1.24i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.746962608428975151290573322701, −8.973470323374918590167986438162, −7.989274500627122250345230383755, −7.55109088992399047227513991285, −6.78794583613904767299617615943, −4.63878102470885938175632249626, −4.09901415201038295276554185606, −3.38625114711668295509669877800, −2.65696957998082793074051999171, −1.80526770693006699596969868250, 0.71130477078367937931328558757, 1.36169309743605122359751698456, 3.60447991794820900570297940651, 4.53004346447199634314622698497, 5.52647019274529824912535224678, 6.20962721505017656047579979260, 6.98744340780661657193555296418, 7.83009109524472000445247188080, 8.387592073048417118494115809888, 8.736137446316887382668196224276

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