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Properties

Label 2-1323-63.25-c1-0-0
Degree $2$
Conductor $1323$
Sign $-0.783 - 0.621i$
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  − 2.17·2-s + 2.73·4-s + (−0.634 + 1.09i)5-s − 1.60·8-s + (1.38 − 2.39i)10-s + (−2.73 − 4.74i)11-s + (−2.37 − 4.10i)13-s − 1.98·16-s + (2.40 − 4.17i)17-s + (2.69 + 4.66i)19-s + (−1.73 + 3.00i)20-s + (5.96 + 10.3i)22-s + (−2.58 + 4.48i)23-s + (1.69 + 2.93i)25-s + (5.16 + 8.94i)26-s + ⋯
L(s)  = 1  − 1.53·2-s + 1.36·4-s + (−0.283 + 0.491i)5-s − 0.566·8-s + (0.436 − 0.755i)10-s + (−0.825 − 1.43i)11-s + (−0.658 − 1.13i)13-s − 0.496·16-s + (0.584 − 1.01i)17-s + (0.617 + 1.06i)19-s + (−0.388 + 0.672i)20-s + (1.27 + 2.20i)22-s + (−0.539 + 0.934i)23-s + (0.339 + 0.587i)25-s + (1.01 + 1.75i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.783 - 0.621i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.783 - 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1391494510\)
\(L(\frac12)\) \(\approx\) \(0.1391494510\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.17T + 2T^{2} \)
5 \( 1 + (0.634 - 1.09i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.73 + 4.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.37 + 4.10i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.40 + 4.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.69 - 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.58 - 4.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.01 - 3.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 + (0.959 + 1.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.94 + 3.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.66 - 2.87i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.15T + 47T^{2} \)
53 \( 1 + (3.57 - 6.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.308T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 4.47T + 67T^{2} \)
71 \( 1 - 1.96T + 71T^{2} \)
73 \( 1 + (5.27 - 9.13i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 9.01T + 79T^{2} \)
83 \( 1 + (5.08 - 8.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.59 + 4.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.48 + 4.30i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.968869378155795722394622503540, −9.160984450351026061273878429264, −8.251897519891937943844832377362, −7.64939313782759311063322012612, −7.24984322673701023541462288928, −5.88024977706097188464787185107, −5.21134990671438476365979889108, −3.43067750708467185130261787766, −2.75465282657300750073503540262, −1.15936592895542230965359302575, 0.10807319263491498684000188865, 1.65577908656164907719436371537, 2.53667845728255725928856894086, 4.33458464136634299725374124745, 4.91488363247658168612910537189, 6.41588078291891890970178333684, 7.20620552073206203594606743225, 7.83465691662075015937774075659, 8.550567261629326799820645159096, 9.336370778460916569581394900574

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