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Properties

Label 2-3240-360.139-c0-0-7
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: $\chi_{3240} (2539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7510232589\)
\(L(\frac12)\) \(\approx\) \(0.7510232589\)
\(L(1)\) 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

−8.480676807611327247294799801035, −8.209373712572557973251502358896, −7.58152837765408716159878823074, −6.34197433296330745668163290648, −5.68749489108613563593076449986, −4.65063956843494897380920665463, −3.62347378956749057768147089426, −2.85468991959902631254159574940, −2.35505118207077451704885693061, −0.63088498262957786877999365299, 1.14257753522792945873190143295, 2.10995009594499634705880901142, 4.02992299739737853962169585471, 4.52720176469718369979864929211, 5.03957871765039858921651706046, 6.10226298989492232937594995415, 7.03238605825532987182737946157, 7.63357820498692989287934724390, 8.003579776703579370758019413072, 8.942633923580969335391689214883

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