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Properties

Label 2-2144-536.227-c0-0-0
Degree $2$
Conductor $2144$
Sign $0.836 + 0.548i$
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  + (−1.21 + 0.782i)3-s + (0.454 − 0.996i)9-s + (1.78 − 0.713i)11-s + (−0.642 − 1.85i)17-s + (−1.56 − 0.149i)19-s + (−0.959 − 0.281i)25-s + (0.0196 + 0.136i)27-s + (−1.61 + 2.26i)33-s + (0.815 + 0.157i)41-s + (1.21 − 1.40i)43-s + (0.981 − 0.189i)49-s + (2.23 + 1.75i)51-s + (2.02 − 1.04i)57-s + (0.452 − 0.132i)59-s + (0.142 − 0.989i)67-s + ⋯
L(s)  = 1  + (−1.21 + 0.782i)3-s + (0.454 − 0.996i)9-s + (1.78 − 0.713i)11-s + (−0.642 − 1.85i)17-s + (−1.56 − 0.149i)19-s + (−0.959 − 0.281i)25-s + (0.0196 + 0.136i)27-s + (−1.61 + 2.26i)33-s + (0.815 + 0.157i)41-s + (1.21 − 1.40i)43-s + (0.981 − 0.189i)49-s + (2.23 + 1.75i)51-s + (2.02 − 1.04i)57-s + (0.452 − 0.132i)59-s + (0.142 − 0.989i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2144\)    =    \(2^{5} \cdot 67\)
Sign: $0.836 + 0.548i$
Analytic conductor: \(1.06999\)
Root analytic conductor: \(1.03440\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2144} (495, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2144,\ (\ :0),\ 0.836 + 0.548i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6974792664\)
\(L(\frac12)\) \(\approx\) \(0.6974792664\)
\(L(1)\) 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 \( 1 + (-0.142 + 0.989i)T \)
good3 \( 1 + (1.21 - 0.782i)T + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (0.959 + 0.281i)T^{2} \)
7 \( 1 + (-0.981 + 0.189i)T^{2} \)
11 \( 1 + (-1.78 + 0.713i)T + (0.723 - 0.690i)T^{2} \)
13 \( 1 + (-0.0475 + 0.998i)T^{2} \)
17 \( 1 + (0.642 + 1.85i)T + (-0.786 + 0.618i)T^{2} \)
19 \( 1 + (1.56 + 0.149i)T + (0.981 + 0.189i)T^{2} \)
23 \( 1 + (0.995 - 0.0950i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.0475 - 0.998i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.815 - 0.157i)T + (0.928 + 0.371i)T^{2} \)
43 \( 1 + (-1.21 + 1.40i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (-0.580 - 0.814i)T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.452 + 0.132i)T + (0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.723 - 0.690i)T^{2} \)
71 \( 1 + (0.786 + 0.618i)T^{2} \)
73 \( 1 + (0.607 + 0.243i)T + (0.723 + 0.690i)T^{2} \)
79 \( 1 + (0.888 - 0.458i)T^{2} \)
83 \( 1 + (0.223 + 0.175i)T + (0.235 + 0.971i)T^{2} \)
89 \( 1 + (-0.975 - 0.627i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.152267709229040783168595050086, −8.822402491072328009348073440686, −7.47256345689344673547219393415, −6.52925008074508994626293223145, −6.12187083113757437470660575943, −5.19971865739975294543341998967, −4.32142572080413103162184613769, −3.82371770402772281404802284899, −2.33042067118009517457543922522, −0.62918353677627522818471046451, 1.32594899934853829172833104366, 2.08994509176357780640089114184, 3.99905353134447905005814701658, 4.33174411906261545399006205011, 5.77558245291979145595312017233, 6.23793211444699175277900929462, 6.74719637875052976151585710754, 7.59150176224438462711815048813, 8.574141206372959790550436133431, 9.280478701545392159627786663718

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