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Properties

Label 2-1323-63.4-c1-0-16
Degree $2$
Conductor $1323$
Sign $-0.0866 - 0.996i$
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  + (−0.863 + 1.49i)2-s + (−0.490 − 0.849i)4-s + 3.51·5-s − 1.75·8-s + (−3.03 + 5.25i)10-s + 6.09·11-s + (−0.560 + 0.970i)13-s + (2.49 − 4.32i)16-s + (−0.601 + 1.04i)17-s + (−1.10 − 1.90i)19-s + (−1.72 − 2.98i)20-s + (−5.25 + 9.10i)22-s + 1.27·23-s + 7.33·25-s + (−0.967 − 1.67i)26-s + ⋯
L(s)  = 1  + (−0.610 + 1.05i)2-s + (−0.245 − 0.424i)4-s + 1.57·5-s − 0.621·8-s + (−0.958 + 1.66i)10-s + 1.83·11-s + (−0.155 + 0.269i)13-s + (0.624 − 1.08i)16-s + (−0.146 + 0.252i)17-s + (−0.252 − 0.438i)19-s + (−0.385 − 0.667i)20-s + (−1.12 + 1.94i)22-s + 0.265·23-s + 1.46·25-s + (−0.189 − 0.328i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0866 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0866 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.0866 - 0.996i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.0866 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.758065897\)
\(L(\frac12)\) \(\approx\) \(1.758065897\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.863 - 1.49i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 3.51T + 5T^{2} \)
11 \( 1 - 6.09T + 11T^{2} \)
13 \( 1 + (0.560 - 0.970i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.601 - 1.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.10 + 1.90i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.27T + 23T^{2} \)
29 \( 1 + (-3.10 - 5.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0942 + 0.163i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.78 + 3.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.68 + 2.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.90 + 3.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.86 + 4.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.16 - 7.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.63 - 9.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.00 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.95 - 6.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + (2.65 - 4.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.60 - 7.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.624 - 1.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.77 + 4.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.24 + 14.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.472216082020033288102035420483, −9.019700535899582307213670509188, −8.461566441200859350265030102780, −6.99685847864640090869563523185, −6.76056285567733782869732913427, −5.96449103379430663125030665560, −5.20951669863518492273963522557, −3.86824352466801887427306175582, −2.49204918381363744872935851064, −1.27815962758266043552203197539, 1.08874899129151877293826739061, 1.88981987401251061355906093772, 2.82680498301503007001333375211, 3.98025586055326831404215356840, 5.27991576337730961715095030474, 6.30918216674234245311233137140, 6.59479065283043842413187542561, 8.197802902563409737910452425325, 9.053387462073202406708165529153, 9.623018673049777530862084610885

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