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Properties

Label 2-1323-63.59-c1-0-23
Degree $2$
Conductor $1323$
Sign $0.916 + 0.400i$
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  + (1.5 + 0.866i)2-s + (0.5 + 0.866i)4-s − 3·5-s − 1.73i·8-s + (−4.5 − 2.59i)10-s + 1.73i·11-s + (1.5 + 0.866i)13-s + (2.49 − 4.33i)16-s + (1.5 − 2.59i)17-s + (4.5 − 2.59i)19-s + (−1.5 − 2.59i)20-s + (−1.49 + 2.59i)22-s − 5.19i·23-s + 4·25-s + (1.5 + 2.59i)26-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + (0.250 + 0.433i)4-s − 1.34·5-s − 0.612i·8-s + (−1.42 − 0.821i)10-s + 0.522i·11-s + (0.416 + 0.240i)13-s + (0.624 − 1.08i)16-s + (0.363 − 0.630i)17-s + (1.03 − 0.596i)19-s + (−0.335 − 0.580i)20-s + (−0.319 + 0.553i)22-s − 1.08i·23-s + 0.800·25-s + (0.294 + 0.509i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (962, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.916 + 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.103700456\)
\(L(\frac12)\) \(\approx\) \(2.103700456\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.5 + 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.19iT - 23T^{2} \)
29 \( 1 + (-4.5 + 2.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.5 - 4.33i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12 + 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (-4.5 - 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.5 + 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.530235682617509298892648856474, −8.585368582330651364934799202274, −7.58601057620460057814861232738, −7.13267098647986541258074519344, −6.25627095719402938198468542649, −5.20035891424611900075407737986, −4.46935596736090048007741153300, −3.81726085036152213733946959978, −2.80766135364615153005046163132, −0.70130375055115347042275108645, 1.34542189728648312782016143691, 3.10836248469462041144498871373, 3.47542406304217755388431594863, 4.35013173237213191048799836224, 5.25921481403445240316861549242, 6.09240038635763858456755089859, 7.33736990755634790394001229971, 8.129543667534550246380302483436, 8.595002802637933244445799229005, 9.932337160153731746957062503742

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