Properties

Label 2-3645-135.14-c0-0-2
Degree $2$
Conductor $3645$
Sign $0.230 + 0.973i$
Analytic cond. $1.81909$
Root an. cond. $1.34873$
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  + (−0.326 − 1.85i)2-s + (−2.37 + 0.866i)4-s + (0.766 + 0.642i)5-s + (1.43 + 2.49i)8-s + (0.939 − 1.62i)10-s + (2.20 − 1.85i)16-s + (−0.173 + 0.300i)17-s + (−0.173 − 0.300i)19-s + (−2.37 − 0.866i)20-s + (−1.43 + 0.524i)23-s + (0.173 + 0.984i)25-s + (1.76 − 0.642i)31-s + (−1.93 − 1.62i)32-s + (0.613 + 0.223i)34-s + (−0.5 + 0.419i)38-s + ⋯
L(s)  = 1  + (−0.326 − 1.85i)2-s + (−2.37 + 0.866i)4-s + (0.766 + 0.642i)5-s + (1.43 + 2.49i)8-s + (0.939 − 1.62i)10-s + (2.20 − 1.85i)16-s + (−0.173 + 0.300i)17-s + (−0.173 − 0.300i)19-s + (−2.37 − 0.866i)20-s + (−1.43 + 0.524i)23-s + (0.173 + 0.984i)25-s + (1.76 − 0.642i)31-s + (−1.93 − 1.62i)32-s + (0.613 + 0.223i)34-s + (−0.5 + 0.419i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3645\)    =    \(3^{6} \cdot 5\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(1.81909\)
Root analytic conductor: \(1.34873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3645} (2834, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3645,\ (\ :0),\ 0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.011648677\)
\(L(\frac12)\) \(\approx\) \(1.011648677\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.766 - 0.642i)T \)
good2 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (-0.173 + 0.984i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
53 \( 1 - 1.53T + T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.173 + 0.984i)T^{2} \)
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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

−8.794794635544691458534582365592, −8.208736672409372398890366521904, −7.25318443636047307597611134456, −6.17993867863070230854920016061, −5.40371039676906504403199385019, −4.31021443371172629316023055738, −3.72890808550084697479666423444, −2.61169357018305344432707520978, −2.25385205013438496565366681181, −1.10495264906303686108661848055, 0.811097466293185967335315788683, 2.25369818063018688770212636884, 3.93721153015069067916793801777, 4.66832609433507449553983366227, 5.36591504470912314451047059121, 6.03595237564324473860447159782, 6.57241418570824096082603694745, 7.34603048394051273406385165147, 8.265741890858837523552086543522, 8.561364466608054065793248401387

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