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Properties

Label 2-1323-63.25-c1-0-27
Degree $2$
Conductor $1323$
Sign $-0.755 + 0.655i$
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  − 2.46·2-s + 4.05·4-s + (1.82 − 3.16i)5-s − 5.05·8-s + (−4.50 + 7.79i)10-s + (0.203 + 0.351i)11-s + (−0.243 − 0.421i)13-s + 4.32·16-s + (−2.42 + 4.20i)17-s + (−0.986 − 1.70i)19-s + (7.41 − 12.8i)20-s + (−0.5 − 0.866i)22-s + (2.32 − 4.02i)23-s + (−4.19 − 7.25i)25-s + (0.598 + 1.03i)26-s + ⋯
L(s)  = 1  − 1.73·2-s + 2.02·4-s + (0.817 − 1.41i)5-s − 1.78·8-s + (−1.42 + 2.46i)10-s + (0.0612 + 0.106i)11-s + (−0.0675 − 0.116i)13-s + 1.08·16-s + (−0.588 + 1.01i)17-s + (−0.226 − 0.392i)19-s + (1.65 − 2.87i)20-s + (−0.106 − 0.184i)22-s + (0.484 − 0.839i)23-s + (−0.838 − 1.45i)25-s + (0.117 + 0.203i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.755 + 0.655i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5651277774\)
\(L(\frac12)\) \(\approx\) \(0.5651277774\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.46T + 2T^{2} \)
5 \( 1 + (-1.82 + 3.16i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.203 - 0.351i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.243 + 0.421i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.42 - 4.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.986 + 1.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.32 + 4.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.82 + 6.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.02T + 31T^{2} \)
37 \( 1 + (1.16 + 2.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.75 + 6.50i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.16 + 2.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.31T + 47T^{2} \)
53 \( 1 + (1.78 - 3.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.11T + 59T^{2} \)
61 \( 1 - 8.02T + 61T^{2} \)
67 \( 1 - 3.60T + 67T^{2} \)
71 \( 1 + 8.46T + 71T^{2} \)
73 \( 1 + (-0.986 + 1.70i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.16T + 79T^{2} \)
83 \( 1 + (6.08 - 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.41 + 12.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.74 - 8.21i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.184433062620140668493778377556, −8.631107646654688792307234413083, −8.176835502305385076290677514402, −7.04594486678233568742716469434, −6.24425928352753514699912582102, −5.29792276352109546703460381098, −4.20675792631025688991338487847, −2.37623066443111690723583729686, −1.58417748683689928144700318186, −0.41282626173249497501780919409, 1.49931259116447002012373247851, 2.48467502629714309032843095202, 3.33134772842876584367018283957, 5.18360878848245287746151197403, 6.35943623882130171861546341056, 6.88849848703584986445332440678, 7.45938187605495455290911758781, 8.491934205997079343626232495561, 9.290188938328988779574991626499, 9.873799823096550113654212999387

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