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Properties

Label 2-2144-536.467-c0-0-0
Degree $2$
Conductor $2144$
Sign $0.751 + 0.659i$
Analytic cond. $1.06999$
Root an. cond. $1.03440$
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  + (−1.50 − 0.442i)3-s + (1.23 + 0.795i)9-s + (−0.0930 − 0.268i)11-s + (−1.15 + 0.110i)17-s + (1.74 + 0.899i)19-s + (−0.142 − 0.989i)25-s + (−0.485 − 0.560i)27-s + (0.0212 + 0.446i)33-s + (0.975 + 1.37i)41-s + (0.271 − 0.595i)43-s + (0.580 − 0.814i)49-s + (1.79 + 0.345i)51-s + (−2.23 − 2.13i)57-s + (0.264 − 1.83i)59-s + (0.654 − 0.755i)67-s + ⋯
L(s)  = 1  + (−1.50 − 0.442i)3-s + (1.23 + 0.795i)9-s + (−0.0930 − 0.268i)11-s + (−1.15 + 0.110i)17-s + (1.74 + 0.899i)19-s + (−0.142 − 0.989i)25-s + (−0.485 − 0.560i)27-s + (0.0212 + 0.446i)33-s + (0.975 + 1.37i)41-s + (0.271 − 0.595i)43-s + (0.580 − 0.814i)49-s + (1.79 + 0.345i)51-s + (−2.23 − 2.13i)57-s + (0.264 − 1.83i)59-s + (0.654 − 0.755i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2144\)    =    \(2^{5} \cdot 67\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(1.06999\)
Root analytic conductor: \(1.03440\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2144} (1807, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2144,\ (\ :0),\ 0.751 + 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6622175231\)
\(L(\frac12)\) \(\approx\) \(0.6622175231\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
67 \( 1 + (-0.654 + 0.755i)T \)
good3 \( 1 + (1.50 + 0.442i)T + (0.841 + 0.540i)T^{2} \)
5 \( 1 + (0.142 + 0.989i)T^{2} \)
7 \( 1 + (-0.580 + 0.814i)T^{2} \)
11 \( 1 + (0.0930 + 0.268i)T + (-0.786 + 0.618i)T^{2} \)
13 \( 1 + (-0.235 + 0.971i)T^{2} \)
17 \( 1 + (1.15 - 0.110i)T + (0.981 - 0.189i)T^{2} \)
19 \( 1 + (-1.74 - 0.899i)T + (0.580 + 0.814i)T^{2} \)
23 \( 1 + (0.888 - 0.458i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.235 - 0.971i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.975 - 1.37i)T + (-0.327 + 0.945i)T^{2} \)
43 \( 1 + (-0.271 + 0.595i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (-0.0475 + 0.998i)T^{2} \)
53 \( 1 + (0.654 - 0.755i)T^{2} \)
59 \( 1 + (-0.264 + 1.83i)T + (-0.959 - 0.281i)T^{2} \)
61 \( 1 + (0.786 + 0.618i)T^{2} \)
71 \( 1 + (-0.981 - 0.189i)T^{2} \)
73 \( 1 + (-0.651 + 1.88i)T + (-0.786 - 0.618i)T^{2} \)
79 \( 1 + (-0.723 - 0.690i)T^{2} \)
83 \( 1 + (-1.28 - 0.247i)T + (0.928 + 0.371i)T^{2} \)
89 \( 1 + (0.0913 - 0.0268i)T + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.333738770520749242654274653967, −8.219183393185730369841316749647, −7.50950285618363506594206595110, −6.61908605227178047143342026900, −6.09746595300163642879016986241, −5.30990694310484075217448892295, −4.61284745388687636055410321883, −3.46873913456180430688603305854, −2.05569021194685120484777151246, −0.76976233827968632880678087439, 0.973120295981649043933342574491, 2.57321145821135734613678591479, 3.88803545688660813017921336710, 4.73207149243086290921009442587, 5.38172237180287654321849302900, 6.02918709564966009147236417401, 7.02955020310670541429828866203, 7.44843835664319860389898814035, 8.845673800711721833466620656639, 9.462055699582620532256313816513

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