Properties

Label 2-1323-63.20-c1-0-27
Degree $2$
Conductor $1323$
Sign $0.996 - 0.0785i$
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  + (1.81 + 1.04i)2-s + (1.19 + 2.07i)4-s + (−1.04 − 1.80i)5-s + 0.819i·8-s − 4.37i·10-s + (2.79 + 1.61i)11-s + (2.68 − 1.55i)13-s + (1.53 − 2.65i)16-s − 1.63·17-s − 5.53i·19-s + (2.49 − 4.32i)20-s + (3.38 + 5.85i)22-s + (−1.00 + 0.580i)23-s + (0.316 − 0.547i)25-s + 6.50·26-s + ⋯
L(s)  = 1  + (1.28 + 0.740i)2-s + (0.597 + 1.03i)4-s + (−0.467 − 0.809i)5-s + 0.289i·8-s − 1.38i·10-s + (0.843 + 0.486i)11-s + (0.745 − 0.430i)13-s + (0.383 − 0.663i)16-s − 0.395·17-s − 1.26i·19-s + (0.558 − 0.967i)20-s + (0.721 + 1.24i)22-s + (−0.209 + 0.121i)23-s + (0.0632 − 0.109i)25-s + 1.27·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.996 - 0.0785i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.996 - 0.0785i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.267373206\)
\(L(\frac12)\) \(\approx\) \(3.267373206\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 - 1.04i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.04 + 1.80i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.79 - 1.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.68 + 1.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.63T + 17T^{2} \)
19 \( 1 + 5.53iT - 19T^{2} \)
23 \( 1 + (1.00 - 0.580i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.05 - 4.07i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.16 + 2.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 + (-1.35 - 2.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.974 - 1.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.06 - 7.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.09iT - 53T^{2} \)
59 \( 1 + (-1.98 - 3.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.15 - 2.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.336 - 0.583i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.01iT - 71T^{2} \)
73 \( 1 + 3.42iT - 73T^{2} \)
79 \( 1 + (-7.07 + 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.54 - 2.67i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.91T + 89T^{2} \)
97 \( 1 + (-2.07 - 1.20i)T + (48.5 + 84.0i)T^{2} \)
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

−9.455549919769501811350931212841, −8.641297129831003835460691554853, −7.903433129539760665926959257138, −6.79516794518888454594725267215, −6.40848629293470785919332669471, −5.23971278539531189086398200343, −4.60203217404072514320779701740, −3.96817620544200676373197101392, −2.86849345932315781563333484321, −1.02395926704506615790178479298, 1.50973412934107604393268675728, 2.75005588624086062968334985899, 3.66458741917810459928386466126, 4.09857548384214043508604061115, 5.25424266293697950720898851367, 6.28988686393633982847492659633, 6.72252523478299845784159123008, 8.076668371504532273953314776976, 8.725422538660232473241418006704, 10.00940874911730984300576254280

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