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Properties

Label 2-1323-63.4-c1-0-20
Degree $2$
Conductor $1323$
Sign $0.172 + 0.985i$
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  + (−0.247 + 0.429i)2-s + (0.877 + 1.51i)4-s − 3.69·5-s − 1.86·8-s + (0.915 − 1.58i)10-s + 0.892·11-s + (−0.598 + 1.03i)13-s + (−1.29 + 2.23i)16-s + (−0.124 + 0.216i)17-s + (−1.40 − 2.43i)19-s + (−3.23 − 5.60i)20-s + (−0.221 + 0.383i)22-s − 2.47·23-s + 8.63·25-s + (−0.296 − 0.513i)26-s + ⋯
L(s)  = 1  + (−0.175 + 0.303i)2-s + (0.438 + 0.759i)4-s − 1.65·5-s − 0.658·8-s + (0.289 − 0.501i)10-s + 0.269·11-s + (−0.165 + 0.287i)13-s + (−0.323 + 0.559i)16-s + (−0.0303 + 0.0525i)17-s + (−0.322 − 0.557i)19-s + (−0.724 − 1.25i)20-s + (−0.0471 + 0.0817i)22-s − 0.516·23-s + 1.72·25-s + (−0.0581 − 0.100i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.172 + 0.985i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.172 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3658339955\)
\(L(\frac12)\) \(\approx\) \(0.3658339955\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.247 - 0.429i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.69T + 5T^{2} \)
11 \( 1 - 0.892T + 11T^{2} \)
13 \( 1 + (0.598 - 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.124 - 0.216i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.40 + 2.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + (2.07 + 3.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.79 - 3.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.36 + 4.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.39 - 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.98 + 8.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.08 + 8.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.94 + 8.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.906 + 1.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.40 + 9.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.514 + 0.891i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 + (-0.915 + 1.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.899 + 1.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.16 - 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.20 + 2.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.52 + 9.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.124853675656099384196576692205, −8.403295014061627006281883434108, −7.892496681925932641691078040520, −7.04153682807501320717754015982, −6.58835781889551403287386045094, −5.14373576955403822483828926156, −3.98413201121515381837288306071, −3.56655419505021388298811996358, −2.28480527407416353473641099712, −0.17273109107289023565981315062, 1.21273580301306149368209862530, 2.68840244572170762736917920629, 3.71045253591614713079734222723, 4.58728227890846827878950420718, 5.66926873909813967955168575626, 6.62237614566331527505907809613, 7.43390678930203551390735446198, 8.162817710544090878082521937414, 8.998735986059400251781060118077, 9.945624115478000789671679311858

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