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Properties

Label 2-1323-63.4-c1-0-29
Degree $2$
Conductor $1323$
Sign $0.580 + 0.814i$
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  + (0.673 − 1.16i)2-s + (0.0923 + 0.160i)4-s + 2.53·5-s + 2.94·8-s + (1.70 − 2.95i)10-s − 0.467·11-s + (2.91 − 5.04i)13-s + (1.79 − 3.11i)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 + 0.106·23-s + 1.41·25-s + (−3.92 − 6.79i)26-s + ⋯
L(s)  = 1  + (0.476 − 0.825i)2-s + (0.0461 + 0.0800i)4-s + 1.13·5-s + 1.04·8-s + (0.539 − 0.934i)10-s − 0.141·11-s + (0.807 − 1.39i)13-s + (0.449 − 0.778i)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 + 0.0221·23-s + 0.282·25-s + (−0.769 − 1.33i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.580 + 0.814i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.018209279\)
\(L(\frac12)\) \(\approx\) \(3.018209279\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.673 + 1.16i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.53T + 5T^{2} \)
11 \( 1 + 0.467T + 11T^{2} \)
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 \( 1 - 0.106T + 23T^{2} \)
29 \( 1 + (-4.39 - 7.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.84 + 6.65i)T + (-15.5 + 26.8i)T^{2} \)
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 \( 1 + (-2.66 + 4.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.358 - 0.620i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.368 + 0.637i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.479 + 0.829i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.81 - 8.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + (5.13 - 8.89i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.31 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
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

−9.798271792278178218531489950286, −8.722467464635592603699372999895, −8.022657802839097781239071990890, −6.95093444248054462945136780930, −5.99716148801876841053270899432, −5.27692154099566363983907861586, −4.18676053613416602737986297921, −3.18482075722025928784951019407, −2.34599141543620026831377464676, −1.29352171036580509115759195581, 1.47726445250335300914774040503, 2.40439969192885239379211965645, 4.04508201086301766559115792359, 4.84385619798518983660869980935, 5.81023759783677655683598199670, 6.32738352841875899384179885665, 6.98674395816991368150569421026, 7.964542937116327960907681162047, 9.044088763491144493680768302103, 9.610804111175804708524846902397

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