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Properties

Label 2-1323-63.38-c1-0-27
Degree $2$
Conductor $1323$
Sign $-0.998 - 0.0549i$
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  − 2.27i·2-s − 3.18·4-s + (0.717 − 1.24i)5-s + 2.69i·8-s + (−2.82 − 1.63i)10-s + (2.80 − 1.61i)11-s + (4.43 − 2.55i)13-s − 0.239·16-s + (0.545 − 0.945i)17-s + (3.88 − 2.24i)19-s + (−2.28 + 3.95i)20-s + (−3.68 − 6.37i)22-s + (3.47 + 2.00i)23-s + (1.47 + 2.54i)25-s + (−5.82 − 10.0i)26-s + ⋯
L(s)  = 1  − 1.60i·2-s − 1.59·4-s + (0.320 − 0.555i)5-s + 0.951i·8-s + (−0.894 − 0.516i)10-s + (0.844 − 0.487i)11-s + (1.22 − 0.709i)13-s − 0.0597·16-s + (0.132 − 0.229i)17-s + (0.891 − 0.514i)19-s + (−0.510 + 0.883i)20-s + (−0.784 − 1.35i)22-s + (0.723 + 0.417i)23-s + (0.294 + 0.509i)25-s + (−1.14 − 1.97i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.998 - 0.0549i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.998 - 0.0549i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.784650699\)
\(L(\frac12)\) \(\approx\) \(1.784650699\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.27iT - 2T^{2} \)
5 \( 1 + (-0.717 + 1.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.80 + 1.61i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.43 + 2.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.545 + 0.945i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.88 + 2.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.47 - 2.00i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.02 + 0.593i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.74iT - 31T^{2} \)
37 \( 1 + (-0.119 - 0.207i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.71 + 6.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.82 - 6.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.22T + 47T^{2} \)
53 \( 1 + (6.07 + 3.50i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.47T + 59T^{2} \)
61 \( 1 - 3.26iT - 61T^{2} \)
67 \( 1 - 0.660T + 67T^{2} \)
71 \( 1 - 3.82iT - 71T^{2} \)
73 \( 1 + (6.33 + 3.65i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.66T + 79T^{2} \)
83 \( 1 + (-5.45 + 9.44i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.84 - 11.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.69 - 1.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.230869929501247762957218256025, −8.964091397571599793749020151247, −7.903039658201612147258726812254, −6.64793086069402116682595763535, −5.59736964723952930961888954229, −4.74133372356493768578152832709, −3.59469844567738372568643715851, −3.07296507375940906317933848041, −1.59357341094939829195209777204, −0.855024342725257582547431005142, 1.56682265205370242267146589943, 3.28000836301836801937426831822, 4.34429375620631472743962584593, 5.25066209557016374672806770062, 6.34241168446674725635357032024, 6.54571642827748374470180229216, 7.40186683403281124072351477495, 8.322796105296226412570699598467, 8.989875020162636382155204516797, 9.702113244725944999487019359801

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