↑

Properties

Label 2-1323-63.41-c1-0-30
Degree $2$
Conductor $1323$
Sign $-0.784 + 0.620i$
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  + (−2.24 + 1.29i)2-s + (2.36 − 4.09i)4-s + (0.626 − 1.08i)5-s + 7.07i·8-s + 3.24i·10-s + (−0.534 + 0.308i)11-s + (−1.06 − 0.613i)13-s + (−4.44 − 7.69i)16-s − 4.43·17-s − 1.90i·19-s + (−2.96 − 5.12i)20-s + (0.799 − 1.38i)22-s + (−4.11 − 2.37i)23-s + (1.71 + 2.97i)25-s + 3.18·26-s + ⋯
L(s)  = 1  + (−1.58 + 0.916i)2-s + (1.18 − 2.04i)4-s + (0.280 − 0.485i)5-s + 2.50i·8-s + 1.02i·10-s + (−0.161 + 0.0929i)11-s + (−0.294 − 0.170i)13-s + (−1.11 − 1.92i)16-s − 1.07·17-s − 0.436i·19-s + (−0.662 − 1.14i)20-s + (0.170 − 0.295i)22-s + (−0.857 − 0.495i)23-s + (0.343 + 0.594i)25-s + 0.624·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.784 + 0.620i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.784 + 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05840042591\)
\(L(\frac12)\) \(\approx\) \(0.05840042591\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.24 - 1.29i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.626 + 1.08i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.534 - 0.308i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.06 + 0.613i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.43T + 17T^{2} \)
19 \( 1 + 1.90iT - 19T^{2} \)
23 \( 1 + (4.11 + 2.37i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.07 + 2.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.14 - 1.24i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.66T + 37T^{2} \)
41 \( 1 + (2.09 - 3.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.24 + 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.80 + 6.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.09iT - 53T^{2} \)
59 \( 1 + (1.78 - 3.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12.5 - 7.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.80 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + 11.4iT - 73T^{2} \)
79 \( 1 + (-2.01 - 3.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.36 + 7.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.62T + 89T^{2} \)
97 \( 1 + (8.76 - 5.06i)T + (48.5 - 84.0i)T^{2} \)
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

−9.008206374809292976675183454840, −8.656162298044618303596788287133, −7.82782737081704964569926194779, −6.97681501254866501862274978171, −6.33296397004627144635978444106, −5.41349633019872370396710455374, −4.49268229682834078474186666169, −2.58285924086575142740243273485, −1.42576119984692971300300434857, −0.04098340021150960897531720775, 1.59188074118473109243497146283, 2.50446034995022287045571663804, 3.38514255682091158942534544378, 4.65044485600594054853936040455, 6.22699970303581566600996551019, 6.90976325048031035364177591488, 7.87178071461643870614284393244, 8.452270693882799911617683556185, 9.332737737442555833145322113171, 9.915839697586750949635789135668

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