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Properties

Label 2-1323-9.7-c1-0-27
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: $\chi_{1323} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.323 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1110807874\)
\(L(\frac12)\) \(\approx\) \(0.1110807874\)
\(L(\frac{3}{2})\) 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

−9.144817462440456515354153516905, −8.420788652282911944901934462059, −7.73503043112030531921864598934, −7.29232370268187200197087702368, −6.16699299441378245884895692251, −5.25149987372851686611539363685, −4.77338647965336760158877463861, −3.39874085447781158131052171147, −1.34792348552670712666888502969, −0.06780165836339429777149863186, 1.51937922270408699996711008512, 2.85325818534619537944679647232, 3.28741836402949218510698960832, 4.28951355064078474410758041443, 5.68974356156016576222876541936, 6.98404108264410281551188676168, 7.58798265365840648534406919206, 8.518099270703825113684178248788, 9.201672020903582372029786868812, 10.22993724682103869077116742111

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