Properties

Label 2-2736-19.12-c0-0-0
Degree $2$
Conductor $2736$
Sign $-0.0977 + 0.995i$
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  − 7-s + (−1.5 − 0.866i)13-s + 19-s + (0.5 − 0.866i)25-s − 1.73i·31-s − 1.73i·37-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)61-s + (−1.5 − 0.866i)67-s + (0.5 + 0.866i)73-s + (1.5 − 0.866i)79-s + (1.5 + 0.866i)91-s + 1.73i·103-s + ⋯
L(s)  = 1  − 7-s + (−1.5 − 0.866i)13-s + 19-s + (0.5 − 0.866i)25-s − 1.73i·31-s − 1.73i·37-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)61-s + (−1.5 − 0.866i)67-s + (0.5 + 0.866i)73-s + (1.5 − 0.866i)79-s + (1.5 + 0.866i)91-s + 1.73i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ -0.0977 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7436150819\)
\(L(\frac12)\) \(\approx\) \(0.7436150819\)
\(L(1)\) 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 \( 1 - T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−8.995895180678542776277115577399, −7.81681491718672206284192146434, −7.43020527753381586090306463394, −6.52321417514903310232637302742, −5.71516146990152639360945575936, −4.99497728007132371328790400023, −3.96229600867309390172410672302, −3.01845737590230745737867451544, −2.29035475215703533085928477681, −0.46855008322279034826699165570, 1.49569383600972657544422535565, 2.83421409205258426244237504865, 3.38553863517960802531946724868, 4.67648375550538696526344388330, 5.17710922816259131839983574731, 6.34826758260039717661456176861, 6.91681211065318024499585023462, 7.53189804481116670919883617897, 8.537485418341376265690210624144, 9.441745388027253689956561372730

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