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Properties

Label 2-1323-49.3-c0-0-0
Degree $2$
Conductor $1323$
Sign $0.991 - 0.127i$
Analytic cond. $0.660263$
Root an. cond. $0.812565$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.733 + 0.680i)4-s + (−0.623 − 0.781i)7-s + (0.880 − 0.702i)13-s + (0.0747 + 0.997i)16-s + (1.35 − 0.781i)19-s + (0.365 + 0.930i)25-s + (0.0747 − 0.997i)28-s + (1.17 + 0.680i)31-s + (−1.07 + 0.997i)37-s + (−1.78 − 0.858i)43-s + (−0.222 + 0.974i)49-s + (1.12 + 0.0841i)52-s + (−0.400 − 0.432i)61-s + (−0.623 + 0.781i)64-s + (−0.0747 + 0.129i)67-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)4-s + (−0.623 − 0.781i)7-s + (0.880 − 0.702i)13-s + (0.0747 + 0.997i)16-s + (1.35 − 0.781i)19-s + (0.365 + 0.930i)25-s + (0.0747 − 0.997i)28-s + (1.17 + 0.680i)31-s + (−1.07 + 0.997i)37-s + (−1.78 − 0.858i)43-s + (−0.222 + 0.974i)49-s + (1.12 + 0.0841i)52-s + (−0.400 − 0.432i)61-s + (−0.623 + 0.781i)64-s + (−0.0747 + 0.129i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.991 - 0.127i$
Analytic conductor: \(0.660263\)
Root analytic conductor: \(0.812565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ 0.991 - 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.260106423\)
\(L(\frac12)\) \(\approx\) \(1.260106423\)
\(L(1)\) 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 \( 1 + (0.623 + 0.781i)T \)
good2 \( 1 + (-0.733 - 0.680i)T^{2} \)
5 \( 1 + (-0.365 - 0.930i)T^{2} \)
11 \( 1 + (0.955 - 0.294i)T^{2} \)
13 \( 1 + (-0.880 + 0.702i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.826 - 0.563i)T^{2} \)
19 \( 1 + (-1.35 + 0.781i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.826 - 0.563i)T^{2} \)
29 \( 1 + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (-1.17 - 0.680i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.07 - 0.997i)T + (0.0747 - 0.997i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (1.78 + 0.858i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.733 + 0.680i)T^{2} \)
53 \( 1 + (0.0747 + 0.997i)T^{2} \)
59 \( 1 + (-0.365 + 0.930i)T^{2} \)
61 \( 1 + (0.400 + 0.432i)T + (-0.0747 + 0.997i)T^{2} \)
67 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (1.81 - 0.712i)T + (0.733 - 0.680i)T^{2} \)
79 \( 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (-0.955 - 0.294i)T^{2} \)
97 \( 1 + 0.298iT - T^{2} \)
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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

−10.04569109191186876984708335828, −8.905156344326128167068474151352, −8.183228139237948269605365858679, −7.22201321995505050494731727603, −6.82085120181319243327938625874, −5.82455703818585928790018210757, −4.69929743660325760896394697117, −3.31245763798243748049238061128, −3.16415375258652697324958799728, −1.37265526782097777890976050624, 1.43202350084029815001157095984, 2.59004354622948420244506161721, 3.56085122671168571030679623545, 4.91239835677245062881537285293, 5.88376550570280356622756745050, 6.34601657366885668247943289540, 7.20162117998934061381679241074, 8.258782185702179007803031448875, 9.106547714821445157449342629093, 9.899833070887172440079570392498

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