↑

Properties

Label 2-1323-9.7-c1-0-6
Degree $2$
Conductor $1323$
Sign $0.384 - 0.923i$
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  + (0.335 − 0.580i)2-s + (0.775 + 1.34i)4-s + (−0.712 − 1.23i)5-s + 2.38·8-s − 0.955·10-s + (−2.46 + 4.27i)11-s + (1.37 + 2.38i)13-s + (−0.752 + 1.30i)16-s − 1.11·17-s − 4.01·19-s + (1.10 − 1.91i)20-s + (1.65 + 2.86i)22-s + (2.71 + 4.70i)23-s + (1.48 − 2.57i)25-s + 1.84·26-s + ⋯
L(s)  = 1  + (0.236 − 0.410i)2-s + (0.387 + 0.671i)4-s + (−0.318 − 0.551i)5-s + 0.841·8-s − 0.302·10-s + (−0.743 + 1.28i)11-s + (0.381 + 0.661i)13-s + (−0.188 + 0.326i)16-s − 0.271·17-s − 0.921·19-s + (0.247 − 0.427i)20-s + (0.352 + 0.610i)22-s + (0.566 + 0.981i)23-s + (0.296 − 0.514i)25-s + 0.362·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.384 - 0.923i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.384 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.645186038\)
\(L(\frac12)\) \(\approx\) \(1.645186038\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.335 + 0.580i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.712 + 1.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.46 - 4.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.37 - 2.38i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.11T + 17T^{2} \)
19 \( 1 + 4.01T + 19T^{2} \)
23 \( 1 + (-2.71 - 4.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.40 - 5.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 + (-0.124 - 0.215i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.498 - 0.863i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.73 + 8.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.820T + 53T^{2} \)
59 \( 1 + (-3.29 - 5.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0376 + 0.0651i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.29 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.0804T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + (-0.922 + 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.23 - 12.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + (2.70 - 4.67i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.877164935800440140419114009708, −8.825924994090312989450597962832, −8.249200598286361675253887610808, −7.22231179456634367481557854980, −6.82041846813418450215306483137, −5.33878258008951981115230596820, −4.49606890773173016546728810225, −3.80145136789915058864260012951, −2.57122135092886520089292312922, −1.62877165905945980773682499324, 0.61263406700521809074430341691, 2.29890081927800024349927594220, 3.27579723789538145307130897518, 4.46840124836815489617294736660, 5.50930244328202541416007319943, 6.13554961692916674291049294456, 6.86759343231768163945233619231, 7.82717910803065838143637317379, 8.441480285720625259357671042229, 9.537298363238312338999590118296

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