Properties

Label 2-2736-19.2-c0-0-0
Degree $2$
Conductor $2736$
Sign $-0.660 + 0.750i$
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  + (−0.766 − 1.32i)7-s + (−0.439 − 0.524i)13-s + (−0.5 − 0.866i)19-s + (−0.766 + 0.642i)25-s + (−1.70 + 0.984i)31-s − 1.96i·37-s + (0.326 − 0.118i)43-s + (−0.673 + 1.16i)49-s + (−1.76 − 0.642i)61-s + (1.26 − 0.223i)67-s + (−1.43 − 1.20i)73-s + (0.826 − 0.984i)79-s + (−0.358 + 0.984i)91-s + (1.70 + 0.300i)97-s + (0.592 + 0.342i)103-s + ⋯
L(s)  = 1  + (−0.766 − 1.32i)7-s + (−0.439 − 0.524i)13-s + (−0.5 − 0.866i)19-s + (−0.766 + 0.642i)25-s + (−1.70 + 0.984i)31-s − 1.96i·37-s + (0.326 − 0.118i)43-s + (−0.673 + 1.16i)49-s + (−1.76 − 0.642i)61-s + (1.26 − 0.223i)67-s + (−1.43 − 1.20i)73-s + (0.826 − 0.984i)79-s + (−0.358 + 0.984i)91-s + (1.70 + 0.300i)97-s + (0.592 + 0.342i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.660 + 0.750i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ -0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6599775418\)
\(L(\frac12)\) \(\approx\) \(0.6599775418\)
\(L(1)\) 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 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.439 + 0.524i)T + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.96iT - T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.826 + 0.984i)T + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \)
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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

−8.959255915821378433759589764454, −7.62164374508824831599831794951, −7.38426794964506957885590175845, −6.55616335197641461448633478146, −5.67503059387493730455249575602, −4.75435580176048061653701780178, −3.83338636893518699278408941860, −3.20508390441640231911417072977, −1.92465077296939244059607431080, −0.39379786947555079377368709751, 1.84994584453421408842044431216, 2.65059519333702155460061645144, 3.64453450186779995115191786747, 4.58293949417983782347736193770, 5.65147729795717811349418866711, 6.09426641729857269900994523933, 6.91201706772517568374390783520, 7.88147117799635762846283696592, 8.578214537759129383679067714468, 9.355491046588216570992748504069

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