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Properties

Label 2-3240-360.139-c0-0-9
Degree $2$
Conductor $3240$
Sign $0.939 + 0.342i$
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 + 0.999·8-s + (0.499 + 0.866i)10-s + (−0.5 + 0.866i)16-s − 1.73i·17-s + 19-s − 0.999·20-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (1.5 + 0.866i)31-s + (−0.499 − 0.866i)32-s + (1.49 + 0.866i)34-s + (−0.5 + 0.866i)38-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·8-s + (0.499 + 0.866i)10-s + (−0.5 + 0.866i)16-s − 1.73i·17-s + 19-s − 0.999·20-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (1.5 + 0.866i)31-s + (−0.499 − 0.866i)32-s + (1.49 + 0.866i)34-s + (−0.5 + 0.866i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.939 + 0.342i$
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: \((2,\ 3240,\ (\ :0),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9923286568\)
\(L(\frac12)\) \(\approx\) \(0.9923286568\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.73iT - T^{2} \)
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 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
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

−8.774639717621336695108187650658, −8.063613244514016110045635227022, −7.35701975827551266653767395344, −6.62366266728923752217915642105, −5.74538172411086929913050377280, −5.07868091854618054292975873012, −4.60308608784609179346475750161, −3.24176508697820678014290675531, −1.87788608412848438253145007340, −0.78878678362419557792982122383, 1.35224497254611265376290807971, 2.33271783431510951357366550408, 3.12133694147074461780703827854, 3.95428718467989718258389281024, 4.84980638737296705692982102553, 6.09157546690649956605185555807, 6.52888064506062083826810214654, 7.83330835762579070760500838464, 7.964293178622560157125849434103, 9.114380424932356413029279956265

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