↑

Properties

Label 2-1323-9.7-c1-0-21
Degree $2$
Conductor $1323$
Sign $0.0644 + 0.997i$
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  + (1.23 − 2.13i)2-s + (−2.02 − 3.51i)4-s + (1.82 + 3.16i)5-s − 5.05·8-s + 9.00·10-s + (0.203 − 0.351i)11-s + (−0.243 − 0.421i)13-s + (−2.16 + 3.74i)16-s + 4.85·17-s + 1.97·19-s + (7.41 − 12.8i)20-s + (−0.5 − 0.866i)22-s + (2.32 + 4.02i)23-s + (−4.19 + 7.25i)25-s − 1.19·26-s + ⋯
L(s)  = 1  + (0.869 − 1.50i)2-s + (−1.01 − 1.75i)4-s + (0.817 + 1.41i)5-s − 1.78·8-s + 2.84·10-s + (0.0612 − 0.106i)11-s + (−0.0675 − 0.116i)13-s + (−0.540 + 0.936i)16-s + 1.17·17-s + 0.452·19-s + (1.65 − 2.87i)20-s + (−0.106 − 0.184i)22-s + (0.484 + 0.839i)23-s + (−0.838 + 1.45i)25-s − 0.234·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0644 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0644 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.0644 + 0.997i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.0644 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.034847672\)
\(L(\frac12)\) \(\approx\) \(3.034847672\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 + 2.13i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.82 - 3.16i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.203 + 0.351i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.243 + 0.421i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.85T + 17T^{2} \)
19 \( 1 - 1.97T + 19T^{2} \)
23 \( 1 + (-2.32 - 4.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.82 + 6.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.51 - 6.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.32T + 37T^{2} \)
41 \( 1 + (3.75 + 6.50i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.16 + 2.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.15 + 5.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.56T + 53T^{2} \)
59 \( 1 + (3.05 + 5.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.01 - 6.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.80 + 3.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.46T + 71T^{2} \)
73 \( 1 + 1.97T + 73T^{2} \)
79 \( 1 + (4.08 - 7.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.08 - 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 + (4.74 - 8.21i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.00517735944946852054455697141, −9.120073193146986672441817894211, −7.71256811968579658688795964606, −6.74505028971465800124429651643, −5.78805904625362573444297997297, −5.18107696026346715444508519124, −3.86363038790990747300265373484, −3.08965919559596933433864444967, −2.45783688330130718072644869211, −1.28147385383388691625179733445, 1.24443576818777770188705864344, 3.03502395574556906621397140748, 4.41684134006912109257318217341, 4.86772558989482928108908413035, 5.70118126633629030130931425906, 6.24556496226572116621059402586, 7.29540750206933017502222003903, 8.087037868093595053887647309125, 8.769723584820112919726911697352, 9.473097699864568422984148955315

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