↑

Properties

Label 2-1320-1320.269-c0-0-4
Degree $2$
Conductor $1320$
Sign $0.935 + 0.352i$
Analytic cond. $0.658765$
Root an. cond. $0.811643$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 0.999·10-s + (0.309 + 0.951i)11-s + 0.999·12-s + (−0.5 − 1.53i)13-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (0.190 − 0.587i)17-s + (−0.809 + 0.587i)18-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 0.999·10-s + (0.309 + 0.951i)11-s + 0.999·12-s + (−0.5 − 1.53i)13-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (0.190 − 0.587i)17-s + (−0.809 + 0.587i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.935 + 0.352i$
Analytic conductor: \(0.658765\)
Root analytic conductor: \(0.811643\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :0),\ 0.935 + 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8627750058\)
\(L(\frac12)\) \(\approx\) \(0.8627750058\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
good7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - 0.618T + T^{2} \)
29 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.712523910606642886758634986639, −8.808848693897269441570290871450, −7.76788066296510291932938895884, −7.45787482787624161202501671232, −6.34629245167756733973215084348, −5.62008737653941704086899440671, −4.97238548978165481624013735909, −4.26524548262639947939241386331, −2.56010496911084688912046214954, −0.806773075165128783155228469131, 1.50807618081447885865681931412, 2.93642835869888534384505270268, 3.74995935694348361809145939742, 4.66232846622021594132288701125, 5.60070395168589470803767377196, 6.38549439470224944010661564326, 7.09403253384077000446237230323, 8.837785230970164383155102195477, 9.183511095774658808466230048085, 10.28643441262510343181251681929

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