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Properties

Label 2-3645-135.104-c0-0-6
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  + (0.939 + 0.342i)2-s + (0.173 − 0.984i)5-s + (−0.499 − 0.866i)8-s + (0.5 − 0.866i)10-s + (−0.173 − 0.984i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.766 − 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.766 − 0.642i)31-s + (0.766 − 0.642i)34-s + (0.173 + 0.984i)38-s + (−0.939 + 0.342i)40-s + (−0.5 − 0.866i)46-s + (1.53 − 1.28i)47-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.173 − 0.984i)5-s + (−0.499 − 0.866i)8-s + (0.5 − 0.866i)10-s + (−0.173 − 0.984i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.766 − 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.766 − 0.642i)31-s + (0.766 − 0.642i)34-s + (0.173 + 0.984i)38-s + (−0.939 + 0.342i)40-s + (−0.5 − 0.866i)46-s + (1.53 − 1.28i)47-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: $\chi_{3645} (1619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3645,\ (\ :0),\ 0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.749373472\)
\(L(\frac12)\) \(\approx\) \(1.749373472\)
\(L(1)\) 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 \( 1 + (-0.173 + 0.984i)T \)
good2 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.939 - 0.342i)T^{2} \)
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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

−8.549064766565170851309974833994, −7.74973932907133942584934794344, −6.96740405448110871931394193234, −5.94936231229480138365758779245, −5.55873974686308667335075022640, −4.83493874406097764959002377095, −4.10432322931937498003178646044, −3.37762823176824652585141770970, −2.06688156982795190444875373392, −0.74330842508823620032946020676, 1.76840038390502273896940685863, 2.77051637224199330263393336784, 3.41266129674094107368156179787, 4.12012722682835764452730413348, 5.05596704782149599322642516965, 5.84201981046362160391332642691, 6.38099735873271990158097828731, 7.46122311847928670237367592430, 7.920527616216865570425610152624, 9.005991920492229339493829908394

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