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Properties

Label 2-1323-63.38-c1-0-20
Degree $2$
Conductor $1323$
Sign $0.648 + 0.761i$
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  − 0.981i·2-s + 1.03·4-s + (−0.940 + 1.62i)5-s − 2.98i·8-s + (1.59 + 0.923i)10-s + (3.54 − 2.04i)11-s + (−3.51 + 2.02i)13-s − 0.852·16-s + (−0.810 + 1.40i)17-s + (7.03 − 4.06i)19-s + (−0.974 + 1.68i)20-s + (−2.00 − 3.47i)22-s + (3.73 + 2.15i)23-s + (0.730 + 1.26i)25-s + (1.99 + 3.44i)26-s + ⋯
L(s)  = 1  − 0.694i·2-s + 0.518·4-s + (−0.420 + 0.728i)5-s − 1.05i·8-s + (0.505 + 0.291i)10-s + (1.06 − 0.616i)11-s + (−0.974 + 0.562i)13-s − 0.213·16-s + (−0.196 + 0.340i)17-s + (1.61 − 0.932i)19-s + (−0.217 + 0.377i)20-s + (−0.427 − 0.740i)22-s + (0.778 + 0.449i)23-s + (0.146 + 0.253i)25-s + (0.390 + 0.676i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.648 + 0.761i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.648 + 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.974667912\)
\(L(\frac12)\) \(\approx\) \(1.974667912\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.981iT - 2T^{2} \)
5 \( 1 + (0.940 - 1.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.54 + 2.04i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.51 - 2.02i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.810 - 1.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.03 + 4.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.73 - 2.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.542 - 0.313i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.27iT - 31T^{2} \)
37 \( 1 + (3.97 + 6.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.912 - 1.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.53 - 6.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.93T + 47T^{2} \)
53 \( 1 + (-7.24 - 4.18i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.17T + 59T^{2} \)
61 \( 1 - 3.74iT - 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (3.28 + 1.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 8.36T + 79T^{2} \)
83 \( 1 + (-4.38 + 7.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.90 - 8.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.4 + 6.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.492639102694508497218046236311, −9.095324833227029461907851392831, −7.60813003156984562575875486036, −7.09805784828326681853755927453, −6.45122624313878803616193013750, −5.28076478382825367335153815239, −3.98749673325619326900216468715, −3.25643933281257639036148254487, −2.38281099625717853234946938646, −1.02267491560891229289135620498, 1.16082854769651675501292078023, 2.54240553098669703584772174970, 3.76530155227642065202607122069, 5.00085148900335951013589015708, 5.41163391859971378091465664723, 6.77344913577189181916306320374, 7.12345561692701624470480992583, 8.058210539514475701493795170632, 8.710808767773363580121494778833, 9.665549261558518996066433009855

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