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Properties

Label 2-1323-63.38-c1-0-28
Degree $2$
Conductor $1323$
Sign $0.856 + 0.516i$
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  + 0.293i·2-s + 1.91·4-s + (1.53 − 2.65i)5-s + 1.15i·8-s + (0.778 + 0.449i)10-s + (3.37 − 1.94i)11-s + (2.02 − 1.17i)13-s + 3.48·16-s + (−1.68 + 2.91i)17-s + (−2.20 + 1.27i)19-s + (2.92 − 5.07i)20-s + (0.572 + 0.991i)22-s + (−2.58 − 1.49i)23-s + (−2.18 − 3.78i)25-s + (0.344 + 0.596i)26-s + ⋯
L(s)  = 1  + 0.207i·2-s + 0.956·4-s + (0.684 − 1.18i)5-s + 0.406i·8-s + (0.246 + 0.142i)10-s + (1.01 − 0.587i)11-s + (0.562 − 0.324i)13-s + 0.872·16-s + (−0.408 + 0.706i)17-s + (−0.506 + 0.292i)19-s + (0.654 − 1.13i)20-s + (0.122 + 0.211i)22-s + (−0.538 − 0.310i)23-s + (−0.436 − 0.756i)25-s + (0.0675 + 0.116i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.856 + 0.516i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.550709678\)
\(L(\frac12)\) \(\approx\) \(2.550709678\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.293iT - 2T^{2} \)
5 \( 1 + (-1.53 + 2.65i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.37 + 1.94i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.02 + 1.17i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.68 - 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.20 - 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.58 + 1.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.67 - 2.12i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.472iT - 31T^{2} \)
37 \( 1 + (3.89 + 6.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.12 - 5.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.06 + 3.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.05T + 47T^{2} \)
53 \( 1 + (4.99 + 2.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.68T + 59T^{2} \)
61 \( 1 + 1.60iT - 61T^{2} \)
67 \( 1 - 1.57T + 67T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + (-0.856 - 0.494i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 9.27T + 79T^{2} \)
83 \( 1 + (-5.49 + 9.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.15 + 3.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.98 + 2.87i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.443284759477692261742306780247, −8.596701397811850762213436589697, −8.200008922040806026180701556501, −6.92515259300297106716713616683, −6.11927382405460813257723253152, −5.68497965788061286972053847160, −4.49796608215564383819483610185, −3.44064172752356305612175144886, −2.03221728577359482784220791057, −1.17441309860181449824071022927, 1.57683730199243693966144821960, 2.44834251491007679153559615781, 3.33847798197727530697149442516, 4.44961114535933624706056220682, 5.89126476871473348049094794434, 6.60090247944582696861935923491, 6.88840335191514529645958814623, 7.916126325285227928897717139567, 9.131625252368110006033268483989, 9.841794955697306097322302013074

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