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Properties

Label 2-3645-135.29-c0-0-3
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  + (−0.173 + 0.984i)2-s + (0.766 − 0.642i)5-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.766 − 0.642i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (0.939 + 0.342i)31-s + (−0.939 + 0.342i)34-s + (0.766 + 0.642i)38-s + (0.173 + 0.984i)40-s + (−0.5 + 0.866i)46-s + (−1.87 + 0.684i)47-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.766 − 0.642i)5-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.766 − 0.642i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (0.939 + 0.342i)31-s + (−0.939 + 0.342i)34-s + (0.766 + 0.642i)38-s + (0.173 + 0.984i)40-s + (−0.5 + 0.866i)46-s + (−1.87 + 0.684i)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: $\chi_{3645} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3645,\ (\ :0),\ 0.230 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.530294419\)
\(L(\frac12)\) \(\approx\) \(1.530294419\)
\(L(1)\) 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 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
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 \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
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 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
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

−8.613479581421156957257486364742, −8.226269203512566491597420551607, −7.33051798552147385720047761529, −6.62232426234596818775759011783, −6.01815804087176865737865236062, −5.24004110440419328970067046082, −4.72426405935862315623885425211, −3.34804170001894437915050768054, −2.41455057050641743911004809742, −1.27545315398917443475644337942, 1.10804703631823423267509270960, 2.07099682339411435032142427070, 2.94774433366296085061769486561, 3.43627472177431147534552245088, 4.69484339025276265151916051614, 5.63059473914572955996070029756, 6.36182360992373209714628340488, 7.01049009752766796173779280548, 7.78484147575480843803383004300, 8.868355546509831695928703388614

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