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Properties

Label 2-3648-456.389-c0-0-0
Degree $2$
Conductor $3648$
Sign $-0.274 - 0.961i$
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  + (−0.642 − 0.766i)3-s + (−0.642 + 1.11i)7-s + (−0.173 + 0.984i)9-s + (0.439 − 0.524i)13-s + (−0.866 − 0.5i)19-s + (1.26 − 0.223i)21-s + (−0.766 − 0.642i)25-s + (0.866 − 0.500i)27-s + (−0.984 + 1.70i)31-s + 1.96i·37-s − 0.684·39-s + (−0.118 + 0.326i)43-s + (−0.326 − 0.565i)49-s + (0.173 + 0.984i)57-s + (−0.233 − 0.642i)61-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)3-s + (−0.642 + 1.11i)7-s + (−0.173 + 0.984i)9-s + (0.439 − 0.524i)13-s + (−0.866 − 0.5i)19-s + (1.26 − 0.223i)21-s + (−0.766 − 0.642i)25-s + (0.866 − 0.500i)27-s + (−0.984 + 1.70i)31-s + 1.96i·37-s − 0.684·39-s + (−0.118 + 0.326i)43-s + (−0.326 − 0.565i)49-s + (0.173 + 0.984i)57-s + (−0.233 − 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $-0.274 - 0.961i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ -0.274 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4281420121\)
\(L(\frac12)\) \(\approx\) \(0.4281420121\)
\(L(1)\) 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 \( 1 + (0.642 + 0.766i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.642 - 1.11i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.984 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.96iT - T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.118 - 0.326i)T + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.233 + 0.642i)T + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (1.50 + 0.266i)T + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (0.984 - 0.826i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
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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

−8.659461988441184097082284096832, −8.348751046813553190792467604306, −7.33706000166914261079673561490, −6.54185080258249645920672429770, −6.07551434279124195822156539827, −5.37163886748070934129933981504, −4.56969153561554610339716589688, −3.25009122501456463618093632288, −2.48476716922840162176747658170, −1.43881524255979826642310482724, 0.26632512070614270974989484201, 1.81156226491768515199137228463, 3.30079018030201488404916865403, 4.05590630280587264616446589239, 4.36523915404975979936828306260, 5.76777254121030990686305546496, 5.99588383110559560695793298247, 7.05828389697905436997621931131, 7.53792836604864035127916944316, 8.730122272957341538236250775050

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