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Properties

Label 2-1323-63.16-c1-0-16
Degree $2$
Conductor $1323$
Sign $0.845 + 0.534i$
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  + (0.0341 + 0.0592i)2-s + (0.997 − 1.72i)4-s − 2.66·5-s + 0.273·8-s + (−0.0910 − 0.157i)10-s + 1.59·11-s + (2.62 + 4.54i)13-s + (−1.98 − 3.43i)16-s + (3.27 + 5.67i)17-s + (0.950 − 1.64i)19-s + (−2.65 + 4.60i)20-s + (0.0546 + 0.0946i)22-s + 3.06·23-s + 2.09·25-s + (−0.179 + 0.311i)26-s + ⋯
L(s)  = 1  + (0.0241 + 0.0418i)2-s + (0.498 − 0.864i)4-s − 1.19·5-s + 0.0965·8-s + (−0.0287 − 0.0498i)10-s + 0.482·11-s + (0.728 + 1.26i)13-s + (−0.496 − 0.859i)16-s + (0.793 + 1.37i)17-s + (0.218 − 0.377i)19-s + (−0.594 + 1.02i)20-s + (0.0116 + 0.0201i)22-s + 0.639·23-s + 0.419·25-s + (−0.0352 + 0.0610i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.845 + 0.534i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.845 + 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.640109256\)
\(L(\frac12)\) \(\approx\) \(1.640109256\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0341 - 0.0592i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.66T + 5T^{2} \)
11 \( 1 - 1.59T + 11T^{2} \)
13 \( 1 + (-2.62 - 4.54i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.27 - 5.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.950 + 1.64i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.06T + 23T^{2} \)
29 \( 1 + (-3.19 + 5.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.35 + 5.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.11 - 3.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.69 + 6.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.63 + 9.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.89 + 3.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.44 - 7.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.44 + 9.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.35 - 2.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.66 + 2.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (-1.09 - 1.90i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.406 + 0.704i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.41 - 5.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.235 - 0.407i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.57 - 4.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.590491722935374370392976248131, −8.701544598694594000072756511575, −7.936275011025320139815347483856, −6.96468237383973852067591415574, −6.38869598795538615299097049876, −5.44284696463070122163883958938, −4.25908111726896379061719782544, −3.66485273873818409081327142751, −2.13789413238212362139134875353, −0.906554667972552629130973757767, 1.04512512531403430290184958501, 3.01927436317378270818520685078, 3.32793914514712299744841496252, 4.41602515600589193993606149047, 5.46166553171221527911677860703, 6.70973431220331903977037522836, 7.33503686326279801528924699138, 8.102302258914820544217395940077, 8.548470628861833493050960260026, 9.687500300124037555687855047895

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