Properties

Label 2-1323-63.16-c1-0-2
Degree $2$
Conductor $1323$
Sign $0.880 - 0.474i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.880 - 0.474i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.880 - 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2358634624\)
\(L(\frac12)\) \(\approx\) \(0.2358634624\)
\(L(\frac{3}{2})\) 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

−9.764792419591640312387910348011, −9.166776592258802777271028951292, −8.165562442077159731533692685250, −7.81208960735465246880410374439, −6.74960819862968928951165153646, −5.06312379694768031819924530038, −4.15232262511722069032555843914, −3.29580445552573768104805453942, −2.44696029002176707734253414659, −1.10327951750280438663869569257, 0.16221893030479343460366099650, 1.76028931475236001170397513945, 3.89227568146268560568049834356, 4.69631760313283328104129012846, 5.70959100722992944345412054087, 6.51900334918083076383527889062, 7.22683411042739309695997496032, 7.83995868395391111487377015914, 8.672058553857247378806950843186, 9.203870248299821737910318433164

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