Properties

Label 2-1323-9.4-c1-0-31
Degree $2$
Conductor $1323$
Sign $-0.323 - 0.946i$
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  + (−1.19 − 2.06i)2-s + (−1.84 + 3.20i)4-s + (1.46 − 2.52i)5-s + 4.05·8-s − 6.97·10-s + (−0.676 − 1.17i)11-s + (0.733 − 1.26i)13-s + (−1.13 − 1.96i)16-s − 3.31·17-s − 2.20·19-s + (5.39 + 9.35i)20-s + (−1.61 + 2.79i)22-s + (1.31 − 2.27i)23-s + (−1.76 − 3.05i)25-s − 3.49·26-s + ⋯
L(s)  = 1  + (−0.843 − 1.46i)2-s + (−0.924 + 1.60i)4-s + (0.653 − 1.13i)5-s + 1.43·8-s − 2.20·10-s + (−0.204 − 0.353i)11-s + (0.203 − 0.352i)13-s + (−0.284 − 0.492i)16-s − 0.802·17-s − 0.506·19-s + (1.20 + 2.09i)20-s + (−0.344 + 0.596i)22-s + (0.274 − 0.474i)23-s + (−0.353 − 0.611i)25-s − 0.686·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.323 - 0.946i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5721915741\)
\(L(\frac12)\) \(\approx\) \(0.5721915741\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 2.06i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.46 + 2.52i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.676 + 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.733 + 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.31T + 17T^{2} \)
19 \( 1 + 2.20T + 19T^{2} \)
23 \( 1 + (-1.31 + 2.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.521 + 0.903i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.63 + 2.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + (0.904 - 1.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.17 + 3.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.98 + 3.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.45T + 53T^{2} \)
59 \( 1 + (-6.10 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.279 - 0.484i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.40 - 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + (0.383 + 0.664i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.983 + 1.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.40T + 89T^{2} \)
97 \( 1 + (-4.14 - 7.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.046796227466455896948800613164, −8.694619936572947928736231792116, −8.007743282996066949668649393609, −6.62782478933207611414408113381, −5.50568515531254425849008258991, −4.58830082777263733040026613514, −3.53660210285634723103077047702, −2.38473912817936160507814042443, −1.50306072337415781144239006228, −0.31204070263412432918854034009, 1.77990857596007214772214979914, 3.10012791874377513048963430679, 4.60960352772320597966327625306, 5.59350081695350896790500670038, 6.44781520396583281612011045964, 6.85704723561580915235111582522, 7.57277968066438950226485529654, 8.578975123972315137040873803950, 9.165324977343064058011717914342, 10.05299285763376047242890427327

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