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Properties

Label 2-3645-135.14-c0-0-1
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: $\chi_{3645} (2834, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3645,\ (\ :0),\ -0.230 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.319605918\)
\(L(\frac12)\) \(\approx\) \(1.319605918\)
\(L(1)\) 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

−8.579496642142696082179946638142, −8.038933313102685840631632451928, −7.53523803557178324032535801589, −6.74452797543492343788559316282, −5.87542447470779119470381053979, −5.43055839240802972604119706458, −4.34898727848242622195522475595, −3.90032189161582702662827503091, −2.50627194864857714700282749014, −1.34595772554037149266735576167, 0.77014015216994414075940004411, 2.33101783473370229030080001275, 2.77139293116896321575422762839, 3.77420088841146591466932320896, 4.31750946943726764677477736109, 5.30485592085633938626254478723, 6.54108918756878526537999998708, 7.02075148659398175393574135228, 7.66307921960416278433821479135, 8.553444606826684461377066928423

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