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Properties

Label 2-1323-63.4-c1-0-1
Degree $2$
Conductor $1323$
Sign $0.833 - 0.552i$
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  + (1.23 − 2.13i)2-s + (−2.02 − 3.51i)4-s − 3.65·5-s − 5.05·8-s + (−4.50 + 7.79i)10-s − 0.406·11-s + (−0.243 + 0.421i)13-s + (−2.16 + 3.74i)16-s + (−2.42 + 4.20i)17-s + (−0.986 − 1.70i)19-s + (7.41 + 12.8i)20-s + (−0.5 + 0.866i)22-s − 4.64·23-s + 8.38·25-s + (0.598 + 1.03i)26-s + ⋯
L(s)  = 1  + (0.869 − 1.50i)2-s + (−1.01 − 1.75i)4-s − 1.63·5-s − 1.78·8-s + (−1.42 + 2.46i)10-s − 0.122·11-s + (−0.0675 + 0.116i)13-s + (−0.540 + 0.936i)16-s + (−0.588 + 1.01i)17-s + (−0.226 − 0.392i)19-s + (1.65 + 2.87i)20-s + (−0.106 + 0.184i)22-s − 0.969·23-s + 1.67·25-s + (0.117 + 0.203i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.833 - 0.552i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.833 - 0.552i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3104921258\)
\(L(\frac12)\) \(\approx\) \(0.3104921258\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 + 2.13i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.65T + 5T^{2} \)
11 \( 1 + 0.406T + 11T^{2} \)
13 \( 1 + (0.243 - 0.421i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.42 - 4.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.986 + 1.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.64T + 23T^{2} \)
29 \( 1 + (-3.82 - 6.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.51 - 6.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.16 + 2.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.75 - 6.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.16 - 2.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.15 + 5.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.78 - 3.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.05 + 5.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.01 - 6.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.80 + 3.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.46T + 71T^{2} \)
73 \( 1 + (-0.986 + 1.70i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.08 - 7.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.08 + 10.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.41 + 12.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.74 + 8.21i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.23464388016588096266690760704, −8.932730539708490143892432996985, −8.304706031421910349160120162365, −7.27508460344483878605999333470, −6.22360602863165284599650948486, −4.92129942051083468683416917380, −4.34791324523638748301627546160, −3.60048907939375058519131024649, −2.80495227076343575108738673590, −1.44252647681626216748853835202, 0.099133898907884163371417469189, 2.83920383859375930781129306478, 4.06238260816132558459079462449, 4.33068239787580139932581992055, 5.36819624674403894279085266400, 6.34988801613951772965278374470, 7.08556688324490984685822059020, 7.896605832407161440735873466282, 8.139052489537630850541230421559, 9.143363681883374537508562999731

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