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Properties

Label 2-1323-63.41-c1-0-28
Degree $2$
Conductor $1323$
Sign $0.684 + 0.728i$
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  + (−1.61 + 0.934i)2-s + (0.746 − 1.29i)4-s + (1.25 − 2.17i)5-s − 0.947i·8-s + 4.68i·10-s + (4.85 − 2.80i)11-s + (−0.384 − 0.221i)13-s + (2.37 + 4.11i)16-s + 3.07·17-s − 2.57i·19-s + (−1.87 − 3.23i)20-s + (−5.24 + 9.07i)22-s + (−6.83 − 3.94i)23-s + (−0.639 − 1.10i)25-s + 0.829·26-s + ⋯
L(s)  = 1  + (−1.14 + 0.660i)2-s + (0.373 − 0.646i)4-s + (0.560 − 0.970i)5-s − 0.335i·8-s + 1.48i·10-s + (1.46 − 0.845i)11-s + (−0.106 − 0.0615i)13-s + (0.594 + 1.02i)16-s + 0.746·17-s − 0.590i·19-s + (−0.418 − 0.724i)20-s + (−1.11 + 1.93i)22-s + (−1.42 − 0.822i)23-s + (−0.127 − 0.221i)25-s + 0.162·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.684 + 0.728i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.684 + 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9734679785\)
\(L(\frac12)\) \(\approx\) \(0.9734679785\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 - 0.934i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.25 + 2.17i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.85 + 2.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.384 + 0.221i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.07T + 17T^{2} \)
19 \( 1 + 2.57iT - 19T^{2} \)
23 \( 1 + (6.83 + 3.94i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.71 - 1.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9.06 - 5.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 + (-1.64 + 2.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.75 + 8.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.07 + 1.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.85iT - 53T^{2} \)
59 \( 1 + (3.65 - 6.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.40 - 4.27i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.934 + 1.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.95iT - 71T^{2} \)
73 \( 1 + 8.51iT - 73T^{2} \)
79 \( 1 + (-0.287 - 0.497i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.23 - 7.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.57T + 89T^{2} \)
97 \( 1 + (-3.22 + 1.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.310640280415941673936784522648, −8.648708515388793484670452078075, −8.277542592314970743945651771564, −7.13182477412747239575726002855, −6.36557045713280468989755079462, −5.66333937582382934958136758791, −4.48938818133242921013544934050, −3.42917842950407545467685551345, −1.63340340491204558943371201584, −0.66572912329946348904657481316, 1.36803383652258497707353671377, 2.18717060511418460369999929704, 3.29811846131676910462886739913, 4.44310010236285511946235255067, 5.87474134672273947919790702182, 6.49215406940836758809989464595, 7.56645878373381690064841954758, 8.164152426613100877728050199383, 9.352780233441224755011858045019, 9.830815298938121837599200361648

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