Properties

Degree $2$
Conductor $1323$
Sign $-0.797 + 0.603i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.849 − 1.47i)2-s + (−0.444 − 0.769i)4-s − 3.58·5-s + 1.88·8-s + (−3.04 + 5.28i)10-s + 2.81·11-s + (0.5 − 0.866i)13-s + (2.49 − 4.31i)16-s + (2.05 − 3.56i)17-s + (−0.444 − 0.769i)19-s + (1.59 + 2.76i)20-s + (2.38 − 4.13i)22-s − 5.87·23-s + 7.87·25-s + (−0.849 − 1.47i)26-s + ⋯
L(s)  = 1  + (0.600 − 1.04i)2-s + (−0.222 − 0.384i)4-s − 1.60·5-s + 0.667·8-s + (−0.964 + 1.67i)10-s + 0.847·11-s + (0.138 − 0.240i)13-s + (0.623 − 1.07i)16-s + (0.498 − 0.863i)17-s + (−0.101 − 0.176i)19-s + (0.356 + 0.617i)20-s + (0.509 − 0.882i)22-s − 1.22·23-s + 1.57·25-s + (−0.166 − 0.288i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.797 + 0.603i$
Motivic weight: \(1\)
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.797 + 0.603i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.608537160\)
\(L(\frac12)\) \(\approx\) \(1.608537160\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.849 + 1.47i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.58T + 5T^{2} \)
11 \( 1 - 2.81T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.05 + 3.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.444 + 0.769i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.87T + 23T^{2} \)
29 \( 1 + (0.849 + 1.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.49 + 6.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.38 + 4.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.70 + 4.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.60 + 4.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.33 + 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0618 - 0.107i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.43 - 7.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.93 + 3.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.15 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.87T + 71T^{2} \)
73 \( 1 + (5.32 - 9.21i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.54 + 6.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.05 - 3.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.80 + 8.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.66 - 6.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.458882140492693959000813327920, −8.458284843108176514441810137063, −7.60115267834933038913135553107, −7.13169563031860743978733049774, −5.74011261458025454623240058258, −4.58764418524973941029405561402, −3.88988073287463422834688528533, −3.38701446524073121807416545304, −2.13297887142701466242833420487, −0.58564802765911944437048301100, 1.46462478634798521006361985841, 3.47961128614014436624210476316, 4.05374605374222600874212608289, 4.83170432643784022425888152312, 5.94989455469312028944740030409, 6.66954560311147759930751963496, 7.43780420068870526149089594595, 8.101072262503535878775889287613, 8.703472674779360458127455106671, 9.982092120345589563840525987973

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