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Properties

Label 2-1323-63.47-c1-0-9
Degree $2$
Conductor $1323$
Sign $0.00947 - 0.999i$
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  + (−0.555 + 0.320i)2-s + (−0.794 + 1.37i)4-s − 2.21·5-s − 2.30i·8-s + (1.22 − 0.709i)10-s − 3.39i·11-s + (1.56 − 0.901i)13-s + (−0.849 − 1.47i)16-s + (2.98 + 5.16i)17-s + (1.42 + 0.822i)19-s + (1.75 − 3.04i)20-s + (1.08 + 1.88i)22-s − 2.37i·23-s − 0.111·25-s + (−0.578 + 1.00i)26-s + ⋯
L(s)  = 1  + (−0.392 + 0.226i)2-s + (−0.397 + 0.687i)4-s − 0.988·5-s − 0.813i·8-s + (0.388 − 0.224i)10-s − 1.02i·11-s + (0.432 − 0.249i)13-s + (−0.212 − 0.367i)16-s + (0.723 + 1.25i)17-s + (0.326 + 0.188i)19-s + (0.392 − 0.680i)20-s + (0.232 + 0.401i)22-s − 0.494i·23-s − 0.0222·25-s + (−0.113 + 0.196i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00947 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00947 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.00947 - 0.999i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.00947 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7976744877\)
\(L(\frac12)\) \(\approx\) \(0.7976744877\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.555 - 0.320i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 2.21T + 5T^{2} \)
11 \( 1 + 3.39iT - 11T^{2} \)
13 \( 1 + (-1.56 + 0.901i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.98 - 5.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.42 - 0.822i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.37iT - 23T^{2} \)
29 \( 1 + (2.44 + 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-9.28 - 5.36i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.849 - 1.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.455 + 0.788i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.96 - 3.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.123 - 0.213i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.82 - 3.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.39 - 9.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.22 + 0.709i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.99 + 6.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + (-0.369 + 0.213i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.49 - 4.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.28 - 7.42i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.26 - 9.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.30 - 3.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.746736603711788025751829193440, −8.654219025719998773431684896893, −8.218114300978115895604423228011, −7.75152195623636558162646290990, −6.67718555646996245479707970930, −5.80780826125157405099312609848, −4.52504534237274128647002432529, −3.68503472179983380377095032585, −3.07389547093506200895876677309, −0.981933301990002622881218842672, 0.51215733528332336437506961011, 1.86858803126071739280048903086, 3.25212100826341306938215177971, 4.40787784431799581992558782842, 5.02990895184286463468979016536, 6.09081007523671825081394368398, 7.20908677478128762467120847874, 7.83296132862681516645156709589, 8.698612839508624752941627824472, 9.658128918566087973157387304811

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