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Properties

Label 2-1323-63.47-c1-0-20
Degree $2$
Conductor $1323$
Sign $0.620 - 0.783i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.02 + 0.589i)2-s + (−0.305 + 0.529i)4-s + 4.33·5-s − 3.07i·8-s + (−4.42 + 2.55i)10-s − 2.16i·11-s + (−2.25 + 1.30i)13-s + (1.20 + 2.08i)16-s + (0.585 + 1.01i)17-s + (2.09 + 1.20i)19-s + (−1.32 + 2.29i)20-s + (1.27 + 2.20i)22-s + 3.65i·23-s + 13.7·25-s + (1.53 − 2.65i)26-s + ⋯
L(s)  = 1  + (−0.721 + 0.416i)2-s + (−0.152 + 0.264i)4-s + 1.93·5-s − 1.08i·8-s + (−1.39 + 0.807i)10-s − 0.651i·11-s + (−0.624 + 0.360i)13-s + (0.300 + 0.520i)16-s + (0.142 + 0.245i)17-s + (0.480 + 0.277i)19-s + (−0.296 + 0.513i)20-s + (0.271 + 0.470i)22-s + 0.761i·23-s + 2.75·25-s + (0.300 − 0.520i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.620 - 0.783i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (656, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.620 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.509611334\)
\(L(\frac12)\) \(\approx\) \(1.509611334\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 - 0.589i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 4.33T + 5T^{2} \)
11 \( 1 + 2.16iT - 11T^{2} \)
13 \( 1 + (2.25 - 1.30i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.585 - 1.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.09 - 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.65iT - 23T^{2} \)
29 \( 1 + (0.589 + 0.340i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.67 - 3.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.55 + 4.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.68 + 6.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.12 - 3.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.57 - 6.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.79 + 1.61i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.91 + 5.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.21 + 3.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.32 - 5.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.95iT - 71T^{2} \)
73 \( 1 + (-10.3 + 5.95i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.87 - 8.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.796 + 1.37i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.04 + 5.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.36 - 1.36i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.686760650745062769011175808787, −9.032133736776246335789780746721, −8.306768338886534997500382659226, −7.28869739812270623692642725634, −6.50861734886356548907343707906, −5.77181121330658170664539336200, −4.91860228771687729009424013881, −3.52300050920185637477389249975, −2.38552112766956470940748987222, −1.14375873824820163713894965761, 0.997007922240852799969265055911, 2.11109304455473689200716609383, 2.73692582687400928289219553276, 4.73650564138559460004597020033, 5.29848802188156513758297295742, 6.14426725017889848651167498524, 6.99995590925729240465493484574, 8.197671361713470577269042924100, 9.028607631967106671272471365749, 9.750585653407130993882126022560

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