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Properties

Label 2-1323-9.4-c1-0-22
Degree $2$
Conductor $1323$
Sign $-0.827 + 0.561i$
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  + (−0.849 − 1.47i)2-s + (−0.444 + 0.769i)4-s + (−0.474 + 0.822i)5-s − 1.88·8-s + 1.61·10-s + (−0.294 − 0.509i)11-s + (2.50 − 4.34i)13-s + (2.49 + 4.31i)16-s + 7.58·17-s − 4.46·19-s + (−0.421 − 0.730i)20-s + (−0.5 + 0.866i)22-s + (1.23 − 2.14i)23-s + (2.04 + 3.54i)25-s − 8.53·26-s + ⋯
L(s)  = 1  + (−0.600 − 1.04i)2-s + (−0.222 + 0.384i)4-s + (−0.212 + 0.367i)5-s − 0.667·8-s + 0.510·10-s + (−0.0886 − 0.153i)11-s + (0.696 − 1.20i)13-s + (0.623 + 1.07i)16-s + 1.83·17-s − 1.02·19-s + (−0.0943 − 0.163i)20-s + (−0.106 + 0.184i)22-s + (0.258 − 0.447i)23-s + (0.409 + 0.709i)25-s − 1.67·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.827 + 0.561i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9961338675\)
\(L(\frac12)\) \(\approx\) \(0.9961338675\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.849 + 1.47i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.474 - 0.822i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.294 + 0.509i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.50 + 4.34i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.58T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
23 \( 1 + (-1.23 + 2.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.73 - 4.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.03 + 5.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.98T + 37T^{2} \)
41 \( 1 + (0.527 - 0.913i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.49 + 6.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.73 + 6.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.92T + 53T^{2} \)
59 \( 1 + (-5.21 + 9.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.82 + 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.93 + 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.30T + 71T^{2} \)
73 \( 1 - 4.46T + 73T^{2} \)
79 \( 1 + (-0.666 - 1.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.84 + 4.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.843T + 89T^{2} \)
97 \( 1 + (-1.70 - 2.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.564741279405152678673248037039, −8.477576897541824679249490526099, −8.095928490057262505568611848023, −6.87916818285397392534092988368, −5.95935714982297031716186297399, −5.10689814802486669447606288779, −3.45497852239459404665767162077, −3.16924564823655312300892475916, −1.78174862329570053816136551702, −0.57000232979116910832636912616, 1.25544276282299860403081880285, 2.92127791181208404222520388550, 4.06609253487023735409247585078, 5.11077746024395299130122751784, 6.12842247572517914364837576857, 6.70406489964247440891066106780, 7.61183893631459353534647550335, 8.341439529904326878345478230581, 8.836959919425296712456386068106, 9.724915107569870093361551304772

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