Properties

Label 2-1323-63.4-c1-0-11
Degree $2$
Conductor $1323$
Sign $0.999 - 0.00483i$
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  + (0.649 − 1.12i)2-s + (0.155 + 0.268i)4-s − 3.52·5-s + 3.00·8-s + (−2.29 + 3.96i)10-s − 1.17·11-s + (1.61 − 2.78i)13-s + (1.64 − 2.84i)16-s + (−2.45 + 4.24i)17-s + (3.43 + 5.94i)19-s + (−0.547 − 0.947i)20-s + (−0.765 + 1.32i)22-s + 4.29·23-s + 7.43·25-s + (−2.09 − 3.62i)26-s + ⋯
L(s)  = 1  + (0.459 − 0.796i)2-s + (0.0775 + 0.134i)4-s − 1.57·5-s + 1.06·8-s + (−0.724 + 1.25i)10-s − 0.355·11-s + (0.446 − 0.773i)13-s + (0.410 − 0.710i)16-s + (−0.594 + 1.02i)17-s + (0.787 + 1.36i)19-s + (−0.122 − 0.211i)20-s + (−0.163 + 0.282i)22-s + 0.896·23-s + 1.48·25-s + (−0.410 − 0.711i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.999 - 0.00483i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.999 - 0.00483i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.709250754\)
\(L(\frac12)\) \(\approx\) \(1.709250754\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.649 + 1.12i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.52T + 5T^{2} \)
11 \( 1 + 1.17T + 11T^{2} \)
13 \( 1 + (-1.61 + 2.78i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.45 - 4.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.43 - 5.94i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.29T + 23T^{2} \)
29 \( 1 + (1.36 + 2.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.960 - 1.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.88 - 8.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.32 + 5.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.83 - 8.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.316 + 0.548i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.11 - 1.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.10 - 7.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.82 + 8.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.66 + 4.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 + (0.519 - 0.898i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.502 - 0.869i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.65 - 6.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.02 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.46 + 9.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.930531325650399020382853549022, −8.574799427101064942674329368690, −7.906513140867660555475965161986, −7.52982566257134515473472767910, −6.33253594977731995614854905632, −5.10638876441370663016418385806, −4.12121130138274026380924700553, −3.58303107799604445834333026166, −2.73411223557144221466315186486, −1.17050497596919874376667206126, 0.73682073725925080440743030347, 2.59689769338737719410722641014, 3.86409846416254428566768021907, 4.63011919400075538505500641891, 5.30262997688614404569485227153, 6.52829823836005419167317016625, 7.28386504037988315634194804666, 7.55185373410359405419944930924, 8.713189349669542667680149582425, 9.381055531947268451431289439399

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