Properties

Label 2-3645-135.119-c0-0-7
Degree $2$
Conductor $3645$
Sign $-0.957 + 0.286i$
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  + (0.266 + 0.223i)2-s + (−0.152 − 0.866i)4-s + (−0.939 − 0.342i)5-s + (0.326 − 0.565i)8-s + (−0.173 − 0.300i)10-s + (−0.613 + 0.223i)16-s + (−0.766 − 1.32i)17-s + (−0.766 + 1.32i)19-s + (−0.152 + 0.866i)20-s + (−0.326 − 1.85i)23-s + (0.766 + 0.642i)25-s + (0.0603 + 0.342i)31-s + (−0.826 − 0.300i)32-s + (0.0923 − 0.524i)34-s + (−0.5 + 0.181i)38-s + ⋯
L(s)  = 1  + (0.266 + 0.223i)2-s + (−0.152 − 0.866i)4-s + (−0.939 − 0.342i)5-s + (0.326 − 0.565i)8-s + (−0.173 − 0.300i)10-s + (−0.613 + 0.223i)16-s + (−0.766 − 1.32i)17-s + (−0.766 + 1.32i)19-s + (−0.152 + 0.866i)20-s + (−0.326 − 1.85i)23-s + (0.766 + 0.642i)25-s + (0.0603 + 0.342i)31-s + (−0.826 − 0.300i)32-s + (0.0923 − 0.524i)34-s + (−0.5 + 0.181i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3645\)    =    \(3^{6} \cdot 5\)
Sign: $-0.957 + 0.286i$
Analytic conductor: \(1.81909\)
Root analytic conductor: \(1.34873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3645} (3239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3645,\ (\ :0),\ -0.957 + 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5153715835\)
\(L(\frac12)\) \(\approx\) \(0.5153715835\)
\(L(1)\) 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 \( 1 + (0.939 + 0.342i)T \)
good2 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.939 + 0.342i)T^{2} \)
11 \( 1 + (-0.766 + 0.642i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + 1.87T + T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.766 + 0.642i)T^{2} \)
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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

−8.348217894558707406186995236900, −7.68294018496709643212251431647, −6.68120448971289959386332942389, −6.30823306611317466438332600102, −5.19350934090736031810383876856, −4.58923133495284016314785593588, −4.04068544403920444247353217594, −2.84842803360831821951496378746, −1.62220228518515193777055090748, −0.26381555023560503528753802150, 1.88161417814101680639694313139, 2.91451894508088047085848293736, 3.68251698740129221176483065740, 4.27285411572349978320348750947, 5.01068837328369366929012488337, 6.21249236122139803940801203826, 6.95892025339076275754778864238, 7.64824673043261237252662421285, 8.270838371491282676899842707776, 8.828544550093099983399114883531

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