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Properties

Label 2-1323-63.25-c1-0-5
Degree $2$
Conductor $1323$
Sign $-0.823 - 0.566i$
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  − 1.34·2-s − 0.184·4-s + (−1.26 + 2.19i)5-s + 2.94·8-s + (1.70 − 2.95i)10-s + (0.233 + 0.405i)11-s + (2.91 + 5.04i)13-s − 3.59·16-s + (−1.93 + 3.35i)17-s + (−1.09 − 1.89i)19-s + (0.233 − 0.405i)20-s + (−0.315 − 0.545i)22-s + (−0.0530 + 0.0918i)23-s + (−0.705 − 1.22i)25-s + (−3.92 − 6.79i)26-s + ⋯
L(s)  = 1  − 0.952·2-s − 0.0923·4-s + (−0.566 + 0.980i)5-s + 1.04·8-s + (0.539 − 0.934i)10-s + (0.0705 + 0.122i)11-s + (0.807 + 1.39i)13-s − 0.899·16-s + (−0.470 + 0.814i)17-s + (−0.250 − 0.434i)19-s + (0.0523 − 0.0906i)20-s + (−0.0672 − 0.116i)22-s + (−0.0110 + 0.0191i)23-s + (−0.141 − 0.244i)25-s + (−0.769 − 1.33i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.823 - 0.566i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.823 - 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5577425314\)
\(L(\frac12)\) \(\approx\) \(0.5577425314\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.34T + 2T^{2} \)
5 \( 1 + (1.26 - 2.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.233 - 0.405i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.91 - 5.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.93 - 3.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 + 1.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0530 - 0.0918i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.39 + 7.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.68T + 31T^{2} \)
37 \( 1 + (-3.84 - 6.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.11 - 1.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.613 - 1.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.33T + 47T^{2} \)
53 \( 1 + (0.358 - 0.620i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.736T + 59T^{2} \)
61 \( 1 + 0.958T + 61T^{2} \)
67 \( 1 + 9.63T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + (5.13 - 8.89i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + (-1.36 + 2.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.05 - 7.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.80 - 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.974458686890385131106484205722, −9.071969107341512586225973865224, −8.390499396459553033444606303060, −7.72964585173547274398376533640, −6.72023385843624936075354553970, −6.29375945383900139638674719230, −4.54988742258254241330735056671, −4.06485363713169926026262677578, −2.71193740472108446055551987836, −1.38951345142649593344204921578, 0.38739031963627469359228648863, 1.30922976347914737338850964670, 3.04225518968553811983517111953, 4.27797714492239535705207081251, 4.93942135405625988814934000333, 5.97319273124277563215589473575, 7.21197716512932420660530394202, 7.999929334081640314493986843501, 8.596492005100514195341095089825, 9.005580565919529751353984203649

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