Properties

Degree $2$
Conductor $3024$
Sign $-0.997 - 0.0709i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.71 + 2.97i)5-s + (0.727 − 2.54i)7-s + (−2.20 − 3.81i)11-s + (1.49 + 2.58i)13-s + (−0.542 + 0.939i)17-s + (3.74 + 6.48i)19-s + (2.16 − 3.74i)23-s + (−3.40 − 5.89i)25-s + (−1.68 + 2.91i)29-s − 9.37·31-s + (6.31 + 6.53i)35-s + (−2.50 − 4.34i)37-s + (1.20 + 2.08i)41-s + (−3.31 + 5.74i)43-s + 3.00·47-s + ⋯
L(s)  = 1  + (−0.768 + 1.33i)5-s + (0.275 − 0.961i)7-s + (−0.664 − 1.15i)11-s + (0.414 + 0.717i)13-s + (−0.131 + 0.227i)17-s + (0.858 + 1.48i)19-s + (0.450 − 0.781i)23-s + (−0.680 − 1.17i)25-s + (−0.312 + 0.541i)29-s − 1.68·31-s + (1.06 + 1.10i)35-s + (−0.412 − 0.714i)37-s + (0.187 + 0.325i)41-s + (−0.505 + 0.875i)43-s + 0.438·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.997 - 0.0709i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.997 - 0.0709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3454071935\)
\(L(\frac12)\) \(\approx\) \(0.3454071935\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.727 + 2.54i)T \)
good5 \( 1 + (1.71 - 2.97i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.20 + 3.81i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.49 - 2.58i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.542 - 0.939i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.74 - 6.48i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.16 + 3.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.68 - 2.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.37T + 31T^{2} \)
37 \( 1 + (2.50 + 4.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.20 - 2.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.31 - 5.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.00T + 47T^{2} \)
53 \( 1 + (-0.530 + 0.919i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 5.42T + 61T^{2} \)
67 \( 1 + 3.33T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (8.21 - 14.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.35T + 79T^{2} \)
83 \( 1 + (1.60 - 2.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.67 + 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.40 - 11.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.936458863370006934800395342994, −8.179917991391487937752139343852, −7.48354190693418198157956377395, −7.03248898964158824372298851853, −6.16071658907863356699555527571, −5.35479427788221822703812694104, −4.05356321725874387971817637259, −3.61905393997567459350084030894, −2.83081276492867890809207841970, −1.42993061595878077928867179309, 0.11151299642544253634098964102, 1.44252970617433034095765317101, 2.55955469319286140943564670782, 3.62008008242845242482798076922, 4.72730612576408861279072774405, 5.13122531044993780041280300366, 5.73996090597144963374623551193, 7.20834557578078189339158436935, 7.57001157308374077601227881792, 8.478393721848519675899054446858

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