Properties

Label 2-1323-63.16-c1-0-23
Degree $2$
Conductor $1323$
Sign $0.717 - 0.696i$
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

Related objects

Downloads

Learn more about

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.717 - 0.696i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.717 - 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.131815894\)
\(L(\frac12)\) \(\approx\) \(3.131815894\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.46 + 9.46i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.881382950593622881667215908231, −8.893003060235969946039315062464, −7.919955458291874519911712228868, −7.10862659291422038692307598434, −6.14373149470809231244101811218, −5.66205308350927998395509061187, −5.13894661025244424491770537528, −3.83035490757413631606946587157, −2.37135867854091888195621158666, −1.46289103517359823037158399684, 1.37575805830661896325088016070, 2.49221807388844841629147815239, 2.93042424105338989363719170714, 4.55677830480132063444701511843, 5.00523528517157168585718678388, 6.20108768877219563491392791414, 6.94689560345105971535413083069, 7.85261260523608790358173044401, 9.110189516013415635192537942728, 9.612652796350414249893839105367

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