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Properties

Label 2-2151-2151.1672-c0-0-12
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: $\chi_{2151} (1672, \cdot )$
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

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

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