↑

Properties

Label 2-1323-9.7-c1-0-22
Degree $2$
Conductor $1323$
Sign $0.972 - 0.234i$
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

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.920 + 1.59i)2-s + (−0.695 − 1.20i)4-s + (0.667 + 1.15i)5-s − 1.12·8-s − 2.45·10-s + (0.756 − 1.31i)11-s + (−2.58 − 4.48i)13-s + (2.42 − 4.19i)16-s + 1.54·17-s + 2.50·19-s + (0.927 − 1.60i)20-s + (1.39 + 2.41i)22-s + (−3.68 − 6.37i)23-s + (1.60 − 2.78i)25-s + 9.53·26-s + ⋯
L(s)  = 1  + (−0.650 + 1.12i)2-s + (−0.347 − 0.601i)4-s + (0.298 + 0.516i)5-s − 0.396·8-s − 0.777·10-s + (0.228 − 0.395i)11-s + (−0.717 − 1.24i)13-s + (0.605 − 1.04i)16-s + 0.375·17-s + 0.574·19-s + (0.207 − 0.359i)20-s + (0.296 + 0.514i)22-s + (−0.767 − 1.32i)23-s + (0.321 − 0.557i)25-s + 1.86·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.972 - 0.234i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.972 - 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9360186054\)
\(L(\frac12)\) \(\approx\) \(0.9360186054\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.920 - 1.59i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.667 - 1.15i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.756 + 1.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.58 + 4.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.54T + 17T^{2} \)
19 \( 1 - 2.50T + 19T^{2} \)
23 \( 1 + (3.68 + 6.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0309 + 0.0536i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.92 - 3.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.563T + 37T^{2} \)
41 \( 1 + (4.51 + 7.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.09 + 8.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.75 - 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.51T + 53T^{2} \)
59 \( 1 + (4.22 + 7.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.61 - 2.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 + 6.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 - 2.75T + 73T^{2} \)
79 \( 1 + (-2.95 + 5.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.80 - 4.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.40T + 89T^{2} \)
97 \( 1 + (6.09 - 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.564345278990750227534976181866, −8.598322133350903274012974487731, −8.013727397903064307471369181752, −7.25391515352014540136251503667, −6.48262521728980365301557650822, −5.79621979815870124946876199695, −4.94237668720372598959236413282, −3.41747466171734404164956121508, −2.51792398163278523428888847970, −0.51905629182881622095124161830, 1.25758442179621031239041700480, 2.03842732221298560253535496416, 3.20437650858647280320172722692, 4.28390274191064982941528267331, 5.30416237954606587029225375068, 6.29341126302335551631423739058, 7.33078731178907457006090007189, 8.244412992290689241993417715265, 9.279521498837825969041309542807, 9.572670141317687827346762093100

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