↑

Properties

Label 2-1323-21.11-c0-0-0
Degree $2$
Conductor $1323$
Sign $0.386 - 0.922i$
Analytic cond. $0.660263$
Root an. cond. $0.812565$
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  + (−0.5 + 0.866i)4-s + 13-s + (−0.499 − 0.866i)16-s + (1 + 1.73i)19-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s − 43-s + (−0.5 + 0.866i)52-s + (−0.5 − 0.866i)61-s + 0.999·64-s + (0.5 − 0.866i)67-s + (1 − 1.73i)73-s − 1.99·76-s + (0.5 + 0.866i)79-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + 13-s + (−0.499 − 0.866i)16-s + (1 + 1.73i)19-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s − 43-s + (−0.5 + 0.866i)52-s + (−0.5 − 0.866i)61-s + 0.999·64-s + (0.5 − 0.866i)67-s + (1 − 1.73i)73-s − 1.99·76-s + (0.5 + 0.866i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(0.660263\)
Root analytic conductor: \(0.812565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (998, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9582765921\)
\(L(\frac12)\) \(\approx\) \(0.9582765921\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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

−9.799398151879063543687478145738, −9.166609446736128091737023508598, −8.203129434453055279785447843626, −7.81544293117752697062202271445, −6.77859874590492802999931802270, −5.77304106386382234872509942124, −4.87748382551209216218171225784, −3.69302898263756161925257416218, −3.27809509975445112864242636305, −1.59269155017624444001947127724, 0.918045495807824505367668021452, 2.34377151969923676383017593319, 3.71286491415362218107883439601, 4.64587814422967234179196333202, 5.50679243439660228228803765276, 6.25406186689216711603220733478, 7.14402998007331631808971790520, 8.208497512958313171949882930359, 9.035839715759652786799637475376, 9.574302464157638868787733535570

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