Properties

Label 2-1323-63.16-c1-0-6
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

\(L(1)\) \(\approx\) \(0.5852492164\)
\(L(\frac12)\) \(\approx\) \(0.5852492164\)
\(L(\frac{3}{2})\) 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

−10.05464094396187451262716024396, −9.006291720849097637331034556054, −8.082886093642741159463082666874, −7.69399047887802155520889706039, −6.16085395317524717262878424907, −5.67388148992940014583711296643, −4.16116522374061159909204771151, −3.39139154140835941680404301797, −2.31801222719081076095649801449, −1.16588928774620046349954382398, 0.32566413189893949728223362840, 2.30482839566329170636242778910, 3.51344545125187962424754222031, 4.68979944488585700893446590713, 5.54826549457739740629359462481, 6.72385333701727512778809335350, 7.08046351754981669883672478986, 7.87830667622589733434736180031, 8.657440133527404543597563398000, 9.472551638081976506183437489024

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