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Properties

Label 2-3240-360.259-c0-0-10
Degree $2$
Conductor $3240$
Sign $-0.173 + 0.984i$
Analytic cond. $1.61697$
Root an. cond. $1.27160$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 0.999·8-s + 0.999·10-s + (1 − 1.73i)11-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 19-s + (0.499 − 0.866i)20-s + (−0.999 − 1.73i)22-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + 0.999·26-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 0.999·8-s + 0.999·10-s + (1 − 1.73i)11-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 19-s + (0.499 − 0.866i)20-s + (−0.999 − 1.73i)22-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + 0.999·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ -0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.804919997\)
\(L(\frac12)\) \(\approx\) \(1.804919997\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 2T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.801166081160006413097852162282, −8.052336699540254873645857124222, −6.75934181824754414442003166572, −6.30395374696762146082758205902, −5.70620898377078290387090232166, −4.36605480183111432885798404319, −3.94559236196324303045140028273, −3.06567604083150391989256361889, −2.05325035246164575603023477542, −1.02918965750779873688277565407, 1.57527254847131054805278285823, 2.54605865091015352993352049611, 3.97443289757123498831039545828, 4.53022784230852668299061356383, 5.31123148712477599616525947491, 5.94346148391479568275326754781, 6.63287889688520218080751389765, 7.59676089410365672855070319435, 8.274814560525591601061423390884, 8.862382048326983612749630284985

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