↑

Properties

Label 2-1323-21.20-c1-0-23
Degree $2$
Conductor $1323$
Sign $-0.654 - 0.755i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + 2.44·5-s + 2.82i·8-s + 3.46i·10-s + 5.65i·11-s + 3.46i·13-s − 4.00·16-s + 2.44·17-s − 1.73i·19-s − 8.00·22-s − 7.07i·23-s + 0.999·25-s − 4.89·26-s + 1.41i·29-s − 5.19i·31-s + ⋯
L(s)  = 1  + 0.999i·2-s + 1.09·5-s + 0.999i·8-s + 1.09i·10-s + 1.70i·11-s + 0.960i·13-s − 1.00·16-s + 0.594·17-s − 0.397i·19-s − 1.70·22-s − 1.47i·23-s + 0.199·25-s − 0.960·26-s + 0.262i·29-s − 0.933i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.654 - 0.755i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1322, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.214549617\)
\(L(\frac12)\) \(\approx\) \(2.214549617\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - 1.41iT - 2T^{2} \)
5 \( 1 - 2.44T + 5T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 2.44T + 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + 7.07iT - 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 2.44T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 - 7.34T + 47T^{2} \)
53 \( 1 - 1.41iT - 53T^{2} \)
59 \( 1 - 14.6T + 59T^{2} \)
61 \( 1 - 5.19iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 9.79T + 83T^{2} \)
89 \( 1 + 2.44T + 89T^{2} \)
97 \( 1 + 8.66iT - 97T^{2} \)
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

−9.845864557982950823005236936671, −9.054497509279055628752783890422, −8.220897700791043483664701886698, −7.09546934070864940502421992157, −6.83319493251373109573172059498, −5.90061519650483049965513452327, −5.09697202808740114855547135786, −4.27648783720070339870586111494, −2.46993964789137036973202544275, −1.83388915671197582229556858611, 0.915507227025875453357793723236, 1.94569743290888129235309167133, 3.14775754718984525975416580052, 3.58292376666547518646566400215, 5.34762215962010658576289169963, 5.81106970216465401383198068022, 6.73160133576794666526167745560, 7.85032912549529423288447586391, 8.737027216024998708475098370479, 9.646142685728487693478572222059

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