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Properties

Label 2-1323-63.4-c1-0-21
Degree $2$
Conductor $1323$
Sign $0.999 + 0.00271i$
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  + (0.335 − 0.580i)2-s + (0.775 + 1.34i)4-s + 1.42·5-s + 2.38·8-s + (0.477 − 0.827i)10-s + 4.93·11-s + (1.37 − 2.38i)13-s + (−0.752 + 1.30i)16-s + (0.559 − 0.969i)17-s + (2.00 + 3.47i)19-s + (1.10 + 1.91i)20-s + (1.65 − 2.86i)22-s − 5.43·23-s − 2.96·25-s + (−0.923 − 1.59i)26-s + ⋯
L(s)  = 1  + (0.236 − 0.410i)2-s + (0.387 + 0.671i)4-s + 0.637·5-s + 0.841·8-s + (0.151 − 0.261i)10-s + 1.48·11-s + (0.381 − 0.661i)13-s + (−0.188 + 0.326i)16-s + (0.135 − 0.235i)17-s + (0.460 + 0.797i)19-s + (0.247 + 0.427i)20-s + (0.352 − 0.610i)22-s − 1.13·23-s − 0.593·25-s + (−0.181 − 0.313i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.999 + 0.00271i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.999 + 0.00271i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.677104077\)
\(L(\frac12)\) \(\approx\) \(2.677104077\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.335 + 0.580i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.42T + 5T^{2} \)
11 \( 1 - 4.93T + 11T^{2} \)
13 \( 1 + (-1.37 + 2.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.559 + 0.969i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.00 - 3.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.43T + 23T^{2} \)
29 \( 1 + (3.40 + 5.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.709 - 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.124 + 0.215i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.73 + 8.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.410 + 0.710i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.29 - 5.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0376 + 0.0651i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.29 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.0804T + 71T^{2} \)
73 \( 1 + (5.34 - 9.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.922 + 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.23 + 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.76 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.70 + 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.844925161840990234668893803722, −8.808462941632873169449822288963, −8.019906151099681064928170297931, −7.19462901195198950104310787749, −6.25297872411504424849723749186, −5.56975668249651666379472639228, −4.12028546827775026432228995737, −3.62148753767076807317296375388, −2.38341613396539103789395400809, −1.39805608919285873414315929541, 1.28173082317204095062584371805, 2.13000516330102814014400462126, 3.70869496775929551179343952389, 4.62816004875467219975905194222, 5.69259041046441647140824258018, 6.29003505452874974154969217351, 6.90098335303163138022409717579, 7.83964017291249991290356841048, 9.116832735721784529775939840168, 9.478926328451590459561479832401

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