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Properties

Label 2-1323-63.16-c1-0-13
Degree $2$
Conductor $1323$
Sign $-0.855 - 0.517i$
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  + (1.08 + 1.88i)2-s + (−1.36 + 2.36i)4-s − 1.26·5-s − 1.60·8-s + (−1.38 − 2.39i)10-s + 5.47·11-s + (2.37 + 4.10i)13-s + (0.992 + 1.71i)16-s + (−2.40 − 4.17i)17-s + (−2.69 + 4.66i)19-s + (1.73 − 3.00i)20-s + (5.96 + 10.3i)22-s + 5.17·23-s − 3.39·25-s + (−5.16 + 8.94i)26-s + ⋯
L(s)  = 1  + (0.769 + 1.33i)2-s + (−0.684 + 1.18i)4-s − 0.567·5-s − 0.566·8-s + (−0.436 − 0.755i)10-s + 1.65·11-s + (0.658 + 1.13i)13-s + (0.248 + 0.429i)16-s + (−0.584 − 1.01i)17-s + (−0.617 + 1.06i)19-s + (0.388 − 0.672i)20-s + (1.27 + 2.20i)22-s + 1.07·23-s − 0.678·25-s + (−1.01 + 1.75i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.855 - 0.517i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.855 - 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.385418723\)
\(L(\frac12)\) \(\approx\) \(2.385418723\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.08 - 1.88i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 1.26T + 5T^{2} \)
11 \( 1 - 5.47T + 11T^{2} \)
13 \( 1 + (-2.37 - 4.10i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.40 + 4.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.69 - 4.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.17T + 23T^{2} \)
29 \( 1 + (2.01 - 3.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.732 + 1.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.959 - 1.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.94 - 3.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.66 - 2.87i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.57 - 2.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.57 + 6.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.154 + 0.267i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.17 + 8.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.23 - 3.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.96T + 71T^{2} \)
73 \( 1 + (-5.27 - 9.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.50 - 7.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.08 + 8.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.59 + 4.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.48 - 4.30i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.602794279947162546857297549079, −8.918062052334808984886422481209, −8.174029852879318222068586024680, −7.20882519931399771699206694371, −6.62214996504930647856877527321, −6.07729154835789366585394026058, −4.86325217126808305168560418611, −4.15672939210226142741111687405, −3.51917242196575267049844526255, −1.58316025980158238757564914231, 0.845794798616424458836368953017, 2.03369191149155971077054945884, 3.27462517772642808641594327257, 3.91926284736683897513808162406, 4.60050867204850978488385985729, 5.76911986200261927517172473888, 6.65685728659939499111163815654, 7.72964434020653319290122780239, 8.774564587697363805270776222599, 9.368341365827961085298295065885

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