Properties

Degree $2$
Conductor $1008$
Sign $0.5 - 0.866i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.592 + 1.62i)3-s + (−0.673 − 1.16i)5-s + (0.5 − 0.866i)7-s + (−2.29 + 1.92i)9-s + (0.826 − 1.43i)11-s + (1.68 + 2.91i)13-s + (1.5 − 1.78i)15-s + 0.467·17-s + 3.22·19-s + (1.70 + 0.300i)21-s + (4.47 + 7.74i)23-s + (1.59 − 2.75i)25-s + (−4.5 − 2.59i)27-s + (−3.13 + 5.42i)29-s + (4.61 + 7.99i)31-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)3-s + (−0.301 − 0.521i)5-s + (0.188 − 0.327i)7-s + (−0.766 + 0.642i)9-s + (0.249 − 0.431i)11-s + (0.467 + 0.809i)13-s + (0.387 − 0.461i)15-s + 0.113·17-s + 0.740·19-s + (0.372 + 0.0656i)21-s + (0.932 + 1.61i)23-s + (0.318 − 0.551i)25-s + (−0.866 − 0.499i)27-s + (−0.582 + 1.00i)29-s + (0.829 + 1.43i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.5 - 0.866i$
Motivic weight: \(1\)
Character: $\chi_{1008} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.752427678\)
\(L(\frac12)\) \(\approx\) \(1.752427678\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.592 - 1.62i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.673 + 1.16i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.826 + 1.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.68 - 2.91i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.467T + 17T^{2} \)
19 \( 1 - 3.22T + 19T^{2} \)
23 \( 1 + (-4.47 - 7.74i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.13 - 5.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.61 - 7.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.23T + 37T^{2} \)
41 \( 1 + (1.70 + 2.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.20 - 3.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.67 + 8.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.573T + 53T^{2} \)
59 \( 1 + (5.19 + 9.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.81 - 6.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.298 - 0.516i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.554T + 71T^{2} \)
73 \( 1 - 2.04T + 73T^{2} \)
79 \( 1 + (1.20 - 2.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.52 - 13.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.08T + 89T^{2} \)
97 \( 1 + (-0.949 + 1.64i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.984869053922974154749324489116, −9.187955173060430422785849399364, −8.662350212946692807922224606391, −7.78748688638382499064473593768, −6.78385447338640196712519830077, −5.52288350965617460420720435936, −4.79418099229561396168914779269, −3.86373547275153675005352696058, −3.06546615981756344861273065414, −1.31806526928197970526707385381, 0.912704431766251368778684771770, 2.42286205617761853709066996569, 3.18910708892034705590491890274, 4.47930790051873363644347740068, 5.80433528669956558524515373121, 6.46379956206924139791086327308, 7.48946892044556602368701499303, 7.939239139855474721399359620038, 8.893685683607703970605930678634, 9.695830157164762560921827237431

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