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Properties

Label 2-1323-9.7-c1-0-7
Degree $2$
Conductor $1323$
Sign $0.512 - 0.858i$
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  + (−0.863 + 1.49i)2-s + (−0.490 − 0.849i)4-s + (−1.75 − 3.04i)5-s − 1.75·8-s + 6.06·10-s + (−3.04 + 5.27i)11-s + (−0.560 − 0.970i)13-s + (2.49 − 4.32i)16-s + 1.20·17-s + 2.20·19-s + (−1.72 + 2.98i)20-s + (−5.25 − 9.10i)22-s + (−0.636 − 1.10i)23-s + (−3.66 + 6.35i)25-s + 1.93·26-s + ⋯
L(s)  = 1  + (−0.610 + 1.05i)2-s + (−0.245 − 0.424i)4-s + (−0.785 − 1.36i)5-s − 0.621·8-s + 1.91·10-s + (−0.918 + 1.59i)11-s + (−0.155 − 0.269i)13-s + (0.624 − 1.08i)16-s + 0.292·17-s + 0.505·19-s + (−0.385 + 0.667i)20-s + (−1.12 − 1.94i)22-s + (−0.132 − 0.229i)23-s + (−0.733 + 1.27i)25-s + 0.379·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.512 - 0.858i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8035830002\)
\(L(\frac12)\) \(\approx\) \(0.8035830002\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.863 - 1.49i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.75 + 3.04i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.04 - 5.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.560 + 0.970i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.20T + 17T^{2} \)
19 \( 1 - 2.20T + 19T^{2} \)
23 \( 1 + (0.636 + 1.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.10 + 5.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.0942 + 0.163i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.57T + 37T^{2} \)
41 \( 1 + (-1.68 - 2.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.90 - 3.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.86 + 4.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.33T + 53T^{2} \)
59 \( 1 + (-5.63 - 9.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.00 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.95 - 6.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 - 5.31T + 73T^{2} \)
79 \( 1 + (4.60 - 7.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.624 + 1.08i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.54T + 89T^{2} \)
97 \( 1 + (8.24 - 14.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.632822397942664702912166094887, −8.661675908820993703072360844030, −8.011831067048481830078216201819, −7.59556262258752656001508834160, −6.76739726299193661769491712407, −5.53103960913619730971803685271, −4.92391215470702337621040256459, −4.00195007537494479221810437014, −2.48227520371625190774512432047, −0.70988051118827600329859957184, 0.68656140408951182924542889979, 2.39359106582897017693875548310, 3.15121942265875305152099372881, 3.71528222131910809302160854533, 5.37215800503306007804471669892, 6.26403499515494072981486792262, 7.19729489514145167426390348298, 8.059087086830261014600139320470, 8.742839445788143674720849231542, 9.771826014759959883521471817658

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