Properties

Label 2-3366-1.1-c1-0-35
Degree $2$
Conductor $3366$
Sign $-1$
Analytic cond. $26.8776$
Root an. cond. $5.18436$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s − 2·7-s − 8-s + 2·10-s + 11-s + 4·13-s + 2·14-s + 16-s − 17-s + 2·19-s − 2·20-s − 22-s − 2·23-s − 25-s − 4·26-s − 2·28-s − 2·29-s − 4·31-s − 32-s + 34-s + 4·35-s + 6·37-s − 2·38-s + 2·40-s − 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s − 0.353·8-s + 0.632·10-s + 0.301·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.458·19-s − 0.447·20-s − 0.213·22-s − 0.417·23-s − 1/5·25-s − 0.784·26-s − 0.377·28-s − 0.371·29-s − 0.718·31-s − 0.176·32-s + 0.171·34-s + 0.676·35-s + 0.986·37-s − 0.324·38-s + 0.316·40-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3366\)    =    \(2 \cdot 3^{2} \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(26.8776\)
Root analytic conductor: \(5.18436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3366} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3366,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
11 \( 1 - T \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−8.314263977249175248487166556648, −7.58325353722904988213090916409, −6.88058020825979319725915000308, −6.18875213971508029398489255414, −5.36978935157106081419405981004, −3.96857785975134090094645009685, −3.63966168110760500189322356889, −2.50328229742078417917317816036, −1.21053447099632842174411855421, 0, 1.21053447099632842174411855421, 2.50328229742078417917317816036, 3.63966168110760500189322356889, 3.96857785975134090094645009685, 5.36978935157106081419405981004, 6.18875213971508029398489255414, 6.88058020825979319725915000308, 7.58325353722904988213090916409, 8.314263977249175248487166556648

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