Properties

Degree $2$
Conductor $1323$
Sign $0.689 - 0.724i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.119 + 0.207i)2-s + (0.971 + 1.68i)4-s + 1.18·5-s − 0.942·8-s + (−0.141 + 0.244i)10-s + 3.70·11-s + (0.5 − 0.866i)13-s + (−1.83 + 3.16i)16-s + (3.47 − 6.01i)17-s + (0.971 + 1.68i)19-s + (1.14 + 1.98i)20-s + (−0.442 + 0.766i)22-s + 5.60·23-s − 3.60·25-s + (0.119 + 0.207i)26-s + ⋯
L(s)  = 1  + (−0.0845 + 0.146i)2-s + (0.485 + 0.841i)4-s + 0.528·5-s − 0.333·8-s + (−0.0446 + 0.0774i)10-s + 1.11·11-s + (0.138 − 0.240i)13-s + (−0.457 + 0.792i)16-s + (0.841 − 1.45i)17-s + (0.222 + 0.385i)19-s + (0.256 + 0.444i)20-s + (−0.0944 + 0.163i)22-s + 1.16·23-s − 0.720·25-s + (0.0234 + 0.0406i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.689 - 0.724i$
Motivic weight: \(1\)
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.689 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.157793090\)
\(L(\frac12)\) \(\approx\) \(2.157793090\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.119 - 0.207i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.18T + 5T^{2} \)
11 \( 1 - 3.70T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.47 + 6.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.971 - 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.60T + 23T^{2} \)
29 \( 1 + (-0.119 - 0.207i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.830 - 1.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.77 - 8.26i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.09 + 8.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.80 - 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.30 + 2.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.80 - 6.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.75 + 3.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.60T + 71T^{2} \)
73 \( 1 + (-7.57 + 13.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.68 - 6.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.47 - 6.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.37 + 2.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.58 - 6.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.455556855391876069046042545916, −9.093113820002831946474066355675, −7.973114628847038412048285822177, −7.32812921393705039679401442158, −6.52548746620881456933013054012, −5.73678439818848612436161337558, −4.60550802267338605211265872712, −3.46229032008104564983169046681, −2.70334208213133594237498182370, −1.29948187193346348284707298411, 1.13028062862217883309692968100, 1.98377374991436671372807473993, 3.27842358194788625540565651967, 4.41942334881816831144023843546, 5.57717554350743648689416266456, 6.17405716946829158014722874082, 6.84653947622334590246696993157, 7.904025922113736426589941486885, 8.993814019723994118169409379518, 9.604120855325566350903445001431

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