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Properties

Label 2-1323-9.7-c1-0-13
Degree $2$
Conductor $1323$
Sign $-0.384 - 0.923i$
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  + (−1.23 + 2.13i)2-s + (−2.02 − 3.51i)4-s + (1.29 + 2.24i)5-s + 5.05·8-s − 6.38·10-s + (2.25 − 3.90i)11-s + (0.5 + 0.866i)13-s + (−2.16 + 3.74i)16-s − 0.945·17-s + 4.05·19-s + (5.25 − 9.10i)20-s + (5.55 + 9.61i)22-s + (−0.136 − 0.236i)23-s + (−0.863 + 1.49i)25-s − 2.46·26-s + ⋯
L(s)  = 1  + (−0.869 + 1.50i)2-s + (−1.01 − 1.75i)4-s + (0.579 + 1.00i)5-s + 1.78·8-s − 2.01·10-s + (0.680 − 1.17i)11-s + (0.138 + 0.240i)13-s + (−0.540 + 0.936i)16-s − 0.229·17-s + 0.930·19-s + (1.17 − 2.03i)20-s + (1.18 + 2.05i)22-s + (−0.0284 − 0.0493i)23-s + (−0.172 + 0.299i)25-s − 0.482·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.384 - 0.923i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.384 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.168780164\)
\(L(\frac12)\) \(\approx\) \(1.168780164\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 2.13i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.29 - 2.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.25 + 3.90i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.945T + 17T^{2} \)
19 \( 1 - 4.05T + 19T^{2} \)
23 \( 1 + (0.136 + 0.236i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.23 + 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.16 - 2.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.78T + 37T^{2} \)
41 \( 1 + (-3.20 - 5.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.21 + 9.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.08 + 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.27T + 53T^{2} \)
59 \( 1 + (-1.36 - 2.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.13 - 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.90 - 13.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 - 1.50T + 73T^{2} \)
79 \( 1 + (7.35 - 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.472 + 0.819i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + (5.74 - 9.95i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.694993657786980261827046769202, −8.856311663567063615940710406959, −8.322440792192456306175364474562, −7.21122822691546863877021714847, −6.78042073605738059619182340599, −5.95473533344781734260162061811, −5.45787903286523831888131345136, −3.97533846128081523065531078710, −2.67597004743822502881920085480, −0.961439780392437005350392772403, 0.954863224303838284389296525243, 1.75403876117183729052527973093, 2.81182589513844942308985658155, 4.04031767573790269977743327848, 4.79899852179538048999484675489, 5.94574331206796356113719102133, 7.27375004670888002501115451128, 8.106746177277276851646332731411, 9.107724516443945570199403713481, 9.369832905892069796392294515300

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