Properties

Label 2-3648-57.23-c0-0-0
Degree $2$
Conductor $3648$
Sign $-0.612 - 0.790i$
Analytic cond. $1.82058$
Root an. cond. $1.34929$
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)3-s + (0.173 − 0.300i)7-s + (−0.939 + 0.342i)9-s + (−0.266 + 1.50i)13-s + (−0.5 + 0.866i)19-s + (0.326 + 0.118i)21-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 1.62i)31-s + 1.87·37-s − 1.53·39-s + (−1.43 + 1.20i)43-s + (0.439 + 0.761i)49-s + (−0.939 − 0.342i)57-s + (−1.17 − 0.984i)61-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)3-s + (0.173 − 0.300i)7-s + (−0.939 + 0.342i)9-s + (−0.266 + 1.50i)13-s + (−0.5 + 0.866i)19-s + (0.326 + 0.118i)21-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 1.62i)31-s + 1.87·37-s − 1.53·39-s + (−1.43 + 1.20i)43-s + (0.439 + 0.761i)49-s + (−0.939 − 0.342i)57-s + (−1.17 − 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $-0.612 - 0.790i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (3329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ -0.612 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.077564551\)
\(L(\frac12)\) \(\approx\) \(1.077564551\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.87T + T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
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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

−9.053825368179503620994259110924, −8.361376034804593202694771043866, −7.65271504012113139768529705218, −6.64030168448643131010264260116, −6.03528575225854966802204188240, −4.92734291840790024920662330494, −4.42448343717046366068550183897, −3.71236625183290344143976766058, −2.71179615606206542573867004447, −1.64901373199787321529360415803, 0.58985391935057391222328414828, 1.94047746366871308628067987901, 2.73702328185048320712100368615, 3.55761877557224468747289257503, 4.80719214729394950553267013823, 5.64478416268704995682875999562, 6.14961110583822505872615969516, 7.21607407207777045919960294633, 7.61227897635776141824831585926, 8.368953822923575840755203396329

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