Properties

Label 2-1323-63.16-c1-0-26
Degree $2$
Conductor $1323$
Sign $-0.947 + 0.318i$
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  + (−0.849 − 1.47i)2-s + (−0.444 + 0.769i)4-s + 0.949·5-s − 1.88·8-s + (−0.806 − 1.39i)10-s + 0.588·11-s + (2.50 + 4.34i)13-s + (2.49 + 4.31i)16-s + (−3.79 − 6.56i)17-s + (2.23 − 3.86i)19-s + (−0.421 + 0.730i)20-s + (−0.5 − 0.866i)22-s − 2.47·23-s − 4.09·25-s + (4.26 − 7.38i)26-s + ⋯
L(s)  = 1  + (−0.600 − 1.04i)2-s + (−0.222 + 0.384i)4-s + 0.424·5-s − 0.667·8-s + (−0.255 − 0.441i)10-s + 0.177·11-s + (0.696 + 1.20i)13-s + (0.623 + 1.07i)16-s + (−0.919 − 1.59i)17-s + (0.511 − 0.886i)19-s + (−0.0943 + 0.163i)20-s + (−0.106 − 0.184i)22-s − 0.516·23-s − 0.819·25-s + (0.836 − 1.44i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.947 + 0.318i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.015589248\)
\(L(\frac12)\) \(\approx\) \(1.015589248\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.849 + 1.47i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 0.949T + 5T^{2} \)
11 \( 1 - 0.588T + 11T^{2} \)
13 \( 1 + (-2.50 - 4.34i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.79 + 6.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.23 + 3.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + (-2.73 + 4.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.03 + 5.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.49 + 6.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.527 + 0.913i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.49 - 6.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.73 + 6.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.46 - 5.99i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.21 + 9.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.82 + 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.93 + 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.30T + 71T^{2} \)
73 \( 1 + (2.23 + 3.86i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.666 - 1.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.84 - 4.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.421 + 0.730i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.70 + 2.94i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.442786723609329292276015481094, −8.913198105618851744975524312545, −7.85408851850932964178573707413, −6.68643179670608819106368932075, −6.11522483204680478898965804548, −4.86842567749190033461537897340, −3.85844486576467001564183492125, −2.62100368993071932562510263900, −1.91545955109300635071564356572, −0.52646847791738160531861880026, 1.41034628489346599728112932949, 2.96037197032706614332103345785, 3.98038287070271854935753017349, 5.41935361804895276663192667432, 6.04409834266890699058559252391, 6.65104908970504469852986498885, 7.68547698417074830017023757222, 8.423054961607872659832446461908, 8.751730398232796180891156588253, 10.04737306599425599703833078065

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