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Properties

Label 2-1323-63.47-c1-0-17
Degree $2$
Conductor $1323$
Sign $0.997 - 0.0716i$
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  + (−1.97 + 1.13i)2-s + (1.59 − 2.75i)4-s − 1.43·5-s + 2.69i·8-s + (2.82 − 1.63i)10-s + 3.23i·11-s + (4.43 − 2.55i)13-s + (0.119 + 0.207i)16-s + (0.545 + 0.945i)17-s + (−3.88 − 2.24i)19-s + (−2.28 + 3.95i)20-s + (−3.68 − 6.37i)22-s − 4.00i·23-s − 2.94·25-s + (−5.82 + 10.0i)26-s + ⋯
L(s)  = 1  + (−1.39 + 0.804i)2-s + (0.795 − 1.37i)4-s − 0.641·5-s + 0.951i·8-s + (0.894 − 0.516i)10-s + 0.975i·11-s + (1.22 − 0.709i)13-s + (0.0298 + 0.0517i)16-s + (0.132 + 0.229i)17-s + (−0.891 − 0.514i)19-s + (−0.510 + 0.883i)20-s + (−0.784 − 1.35i)22-s − 0.835i·23-s − 0.588·25-s + (−1.14 + 1.97i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.997 - 0.0716i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (656, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.997 - 0.0716i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5919227816\)
\(L(\frac12)\) \(\approx\) \(0.5919227816\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.97 - 1.13i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 1.43T + 5T^{2} \)
11 \( 1 - 3.23iT - 11T^{2} \)
13 \( 1 + (-4.43 + 2.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.545 - 0.945i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.88 + 2.24i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.00iT - 23T^{2} \)
29 \( 1 + (1.02 + 0.593i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.24 - 1.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.119 + 0.207i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.71 + 6.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.82 - 6.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.11 - 3.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.07 + 3.50i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.73 + 8.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.82 + 1.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.330 - 0.571i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.82iT - 71T^{2} \)
73 \( 1 + (-6.33 + 3.65i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.83 + 3.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.45 + 9.44i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.84 + 11.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.69 - 1.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.488210671898538322931704570842, −8.516413408367166783307445148650, −8.237598728270184314514918056643, −7.34837068834480556247852820233, −6.62617690046452446163540043062, −5.88197065495549546464994282828, −4.60897097498037202803407275757, −3.56185757817008034075872116320, −1.95073447655321121322462559025, −0.54143477192856033064330157721, 0.901715302209949352100585708674, 2.04920487706899171874631003106, 3.35246443422946048953141656799, 4.01551410622111705747795271728, 5.56898108347127523938353024559, 6.56244978515881523635871230280, 7.58018175391492033727145057608, 8.380321084591771339452448010333, 8.702639409615030636262505304187, 9.616235879374457021933118674647

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