Properties

Label 2-1323-63.16-c1-0-7
Degree $2$
Conductor $1323$
Sign $0.960 + 0.277i$
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  + (−1.35 − 2.35i)2-s + (−2.68 + 4.65i)4-s + 1.58·5-s + 9.15·8-s + (−2.15 − 3.73i)10-s + 1.34·11-s + (1.58 + 2.75i)13-s + (−7.05 − 12.2i)16-s + (1.40 + 2.42i)17-s + (−0.312 + 0.541i)19-s + (−4.26 + 7.38i)20-s + (−1.83 − 3.17i)22-s + 0.284·23-s − 2.48·25-s + (4.31 − 7.47i)26-s + ⋯
L(s)  = 1  + (−0.959 − 1.66i)2-s + (−1.34 + 2.32i)4-s + 0.709·5-s + 3.23·8-s + (−0.681 − 1.17i)10-s + 0.406·11-s + (0.440 + 0.763i)13-s + (−1.76 − 3.05i)16-s + (0.339 + 0.588i)17-s + (−0.0717 + 0.124i)19-s + (−0.952 + 1.65i)20-s + (−0.390 − 0.676i)22-s + 0.0593·23-s − 0.496·25-s + (0.846 − 1.46i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.960 + 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8920918921\)
\(L(\frac12)\) \(\approx\) \(0.8920918921\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 + 2.35i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 1.58T + 5T^{2} \)
11 \( 1 - 1.34T + 11T^{2} \)
13 \( 1 + (-1.58 - 2.75i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.40 - 2.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.312 - 0.541i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.284T + 23T^{2} \)
29 \( 1 + (2.27 - 3.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.71 - 6.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.01 - 6.94i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.01 + 8.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.12 - 5.42i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.57 - 9.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.39 - 2.41i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.28 + 3.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.192 - 0.333i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.26 + 2.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.45T + 71T^{2} \)
73 \( 1 + (-0.234 - 0.405i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.85 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.99 + 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.29 - 2.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.22 + 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.679203654176516501595990018489, −9.009234451627173536322121404098, −8.483000289914366045436945124475, −7.46081821652526103060523490007, −6.43347651516657076783975494620, −5.12074539032295755948056879921, −3.97061386944087834012035992374, −3.24720288693353955682497884711, −1.97943294727964566246097478280, −1.35076763420659789725733554150, 0.55574574804268292425326476360, 1.95644630557693991150651589408, 3.87410497300994343249191007150, 5.19495881851061369890867933968, 5.70281957262685104846156174204, 6.44566165221086187092372190652, 7.28473226141918345630602763822, 7.971828097770756765156841423165, 8.788988187028920748899080208164, 9.465417225957140188804554269171

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