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Properties

Label 2-1323-9.7-c1-0-17
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

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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

\(L(1)\) \(\approx\) \(0.8810933867\)
\(L(\frac12)\) \(\approx\) \(0.8810933867\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.09 + 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.189024140533421308190731619173, −8.698754352486948371513877340849, −8.212816990247333210813317447059, −7.20740247959735024340248268971, −6.43459684169591830839256963850, −5.91176980259914921724132035591, −4.61621463353519774819678001842, −3.83914831279252032556684177045, −2.26997596553377775954855693271, −0.54543191670564138396895654625, 1.06978516568769564252497948577, 2.27849536254003489470884500968, 3.27085955279648949414998814761, 3.99641122451550694050345515880, 5.47793632282458825862598454488, 6.28697334623465182308246483110, 7.38439734084801171310160346428, 8.158085597577711014112263158802, 9.040881607642735893561461260697, 9.699049415860075544631863070198

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