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Properties

Label 2-2736-19.14-c0-0-0
Degree $2$
Conductor $2736$
Sign $0.837 + 0.546i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
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  + (−0.173 − 0.300i)7-s + (1.26 − 0.223i)13-s + (−0.5 − 0.866i)19-s + (−0.173 − 0.984i)25-s + (0.592 − 0.342i)31-s + 0.684i·37-s + (1.43 + 1.20i)43-s + (0.439 − 0.761i)49-s + (−1.17 + 0.984i)61-s + (0.673 − 1.85i)67-s + (0.266 − 1.50i)73-s + (1.93 + 0.342i)79-s + (−0.286 − 0.342i)91-s + (−0.592 − 1.62i)97-s + (1.11 + 0.642i)103-s + ⋯
L(s)  = 1  + (−0.173 − 0.300i)7-s + (1.26 − 0.223i)13-s + (−0.5 − 0.866i)19-s + (−0.173 − 0.984i)25-s + (0.592 − 0.342i)31-s + 0.684i·37-s + (1.43 + 1.20i)43-s + (0.439 − 0.761i)49-s + (−1.17 + 0.984i)61-s + (0.673 − 1.85i)67-s + (0.266 − 1.50i)73-s + (1.93 + 0.342i)79-s + (−0.286 − 0.342i)91-s + (−0.592 − 1.62i)97-s + (1.11 + 0.642i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.837 + 0.546i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ 0.837 + 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.238151335\)
\(L(\frac12)\) \(\approx\) \(1.238151335\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 0.684iT - T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.673 + 1.85i)T + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (-1.93 - 0.342i)T + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \)
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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

−8.908112158284956979818283063001, −8.197303742011951452719986119121, −7.51586210840079767263019905156, −6.40593957024354423929227474105, −6.16742368209472320779461116499, −4.94048917319105621652434294404, −4.19131193662584885211295872997, −3.30504962496349480154310217232, −2.29972583153627645105063306829, −0.927118342063271800821952834942, 1.30633350690278709621312099625, 2.42584975755551579093993924770, 3.57434527778619797696092959446, 4.16317568230237485694964151403, 5.38180426893978093776806788669, 5.97539070574982010648247029506, 6.72745823503081153699784536783, 7.62112511478840128215323505419, 8.404389681883510962042362305674, 9.023253934307894279247950577312

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