↑

Properties

Label 2-1323-63.5-c1-0-27
Degree $2$
Conductor $1323$
Sign $-0.970 + 0.239i$
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  − 1.86i·2-s − 1.49·4-s + (1.25 + 2.17i)5-s − 0.947i·8-s + (4.05 − 2.34i)10-s + (−4.85 − 2.80i)11-s + (−0.384 − 0.221i)13-s − 4.75·16-s + (−1.53 − 2.66i)17-s + (2.22 + 1.28i)19-s + (−1.87 − 3.23i)20-s + (−5.24 + 9.07i)22-s + (6.83 − 3.94i)23-s + (−0.639 + 1.10i)25-s + (−0.414 + 0.718i)26-s + ⋯
L(s)  = 1  − 1.32i·2-s − 0.746·4-s + (0.560 + 0.970i)5-s − 0.335i·8-s + (1.28 − 0.740i)10-s + (−1.46 − 0.845i)11-s + (−0.106 − 0.0615i)13-s − 1.18·16-s + (−0.373 − 0.646i)17-s + (0.511 + 0.295i)19-s + (−0.418 − 0.724i)20-s + (−1.11 + 1.93i)22-s + (1.42 − 0.822i)23-s + (−0.127 + 0.221i)25-s + (−0.0813 + 0.140i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.970 + 0.239i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.970 + 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.358734376\)
\(L(\frac12)\) \(\approx\) \(1.358734376\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.86iT - 2T^{2} \)
5 \( 1 + (-1.25 - 2.17i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.85 + 2.80i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.384 + 0.221i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.53 + 2.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.22 - 1.28i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.83 + 3.94i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.71 - 1.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.4iT - 31T^{2} \)
37 \( 1 + (-0.708 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.64 + 2.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.75 + 8.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.14T + 47T^{2} \)
53 \( 1 + (4.20 - 2.42i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.30T + 59T^{2} \)
61 \( 1 + 8.55iT - 61T^{2} \)
67 \( 1 + 1.86T + 67T^{2} \)
71 \( 1 + 2.95iT - 71T^{2} \)
73 \( 1 + (7.37 - 4.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 0.574T + 79T^{2} \)
83 \( 1 + (-4.23 - 7.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.78 + 6.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.22 + 1.86i)T + (48.5 - 84.0i)T^{2} \)
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

−9.608413215569046752408822819407, −8.740470773900739197541353207773, −7.60059160046844889050174250023, −6.82735994541840869054443313535, −5.83086357618039605387812581670, −4.89892501700679775792707834123, −3.58237875206563780135216829896, −2.75238296852522204533049160543, −2.25719986236606713576147777669, −0.53932406102839885175516026751, 1.58894628663791520114587963384, 2.89156604728945212608848683312, 4.67215501427511224885557689736, 5.10104382793934060268419693729, 5.75202304414236655619521538803, 6.85115744836233817619235138047, 7.46912128926557437423830802393, 8.281749383861622674056191696266, 8.972058906217459466366145059901, 9.697221048639121052447713424630

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