↑

Properties

Label 2-1323-63.20-c1-0-15
Degree $2$
Conductor $1323$
Sign $-0.220 - 0.975i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.220 - 0.975i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.642783488\)
\(L(\frac12)\) \(\approx\) \(3.642783488\)
\(L(\frac{3}{2})\) 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

−10.02527333735762676975139318667, −9.053866699436447616615346130709, −7.87505729895866622656035884190, −6.90626937272417713402896074135, −6.65374999265691761001395925203, −5.68353613088282374733284207269, −4.90805959989148835628723858418, −3.92227515539804276723394782364, −3.13808779889020975340504026859, −1.82668429366951695290917973168, 1.07035084401307856404532069257, 2.32589571629954348815105822733, 3.26025164085007166841613895487, 4.32345760563327852013374116250, 4.99710662773931270517697819832, 5.72040332697659077459486347975, 6.58003512894126945490844472340, 7.77898158912751168887212832047, 8.811926306044111704159179376530, 9.430002471564280728668115765859

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