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Properties

Label 2-1323-9.7-c1-0-20
Degree $2$
Conductor $1323$
Sign $0.766 + 0.642i$
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  + (0.5 − 0.866i)2-s + (0.500 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 3·8-s − 0.999·10-s + (2.5 − 4.33i)11-s + (2.5 + 4.33i)13-s + (0.500 − 0.866i)16-s − 3·17-s + 19-s + (0.499 − 0.866i)20-s + (−2.5 − 4.33i)22-s + (1.5 + 2.59i)23-s + (2 − 3.46i)25-s + 5·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + (−0.223 − 0.387i)5-s + 1.06·8-s − 0.316·10-s + (0.753 − 1.30i)11-s + (0.693 + 1.20i)13-s + (0.125 − 0.216i)16-s − 0.727·17-s + 0.229·19-s + (0.111 − 0.193i)20-s + (−0.533 − 0.923i)22-s + (0.312 + 0.541i)23-s + (0.400 − 0.692i)25-s + 0.980·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.766 + 0.642i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.400454052\)
\(L(\frac12)\) \(\approx\) \(2.400454052\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13T + 89T^{2} \)
97 \( 1 + (-4.5 + 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.425299576344295973716770389234, −8.716782766916085427578498064222, −8.123691234363019259183943381850, −6.94763899406371646143018562965, −6.36265458883412573163699775731, −5.12198921335811558057689442127, −4.04459688183498998307446129871, −3.59274358122106589588075045065, −2.34070684351117564245518225379, −1.13297985530683275244795338550, 1.23422931124403506956313915831, 2.53259380379067120516362596354, 3.85972035716444058078916352829, 4.74613736808218236622065223945, 5.61560984657135122140564774575, 6.53391051084245554000618588398, 7.07778076857527006054714210820, 7.83496120326585387631539143519, 8.857317096816128570703841642266, 9.827786587342908299610877344975

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