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Properties

Label 2-1008-252.103-c1-0-6
Degree $2$
Conductor $1008$
Sign $-0.978 - 0.207i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
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  + (1.71 + 0.266i)3-s + (−2.07 + 1.19i)5-s + (−2.21 + 1.44i)7-s + (2.85 + 0.912i)9-s + (−1.02 − 0.589i)11-s + (−3.94 − 2.27i)13-s + (−3.87 + 1.49i)15-s + (−3.59 + 2.07i)17-s + (0.422 − 0.731i)19-s + (−4.18 + 1.87i)21-s + (−2.61 + 1.50i)23-s + (0.370 − 0.642i)25-s + (4.64 + 2.32i)27-s + (1.38 + 2.39i)29-s − 6.95·31-s + ⋯
L(s)  = 1  + (0.988 + 0.153i)3-s + (−0.928 + 0.535i)5-s + (−0.838 + 0.544i)7-s + (0.952 + 0.304i)9-s + (−0.307 − 0.177i)11-s + (−1.09 − 0.631i)13-s + (−0.999 + 0.386i)15-s + (−0.873 + 0.504i)17-s + (0.0969 − 0.167i)19-s + (−0.912 + 0.409i)21-s + (−0.544 + 0.314i)23-s + (0.0741 − 0.128i)25-s + (0.894 + 0.447i)27-s + (0.257 + 0.445i)29-s − 1.24·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.978 - 0.207i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.978 - 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5808434272\)
\(L(\frac12)\) \(\approx\) \(0.5808434272\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.266i)T \)
7 \( 1 + (2.21 - 1.44i)T \)
good5 \( 1 + (2.07 - 1.19i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.02 + 0.589i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.94 + 2.27i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.59 - 2.07i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.422 + 0.731i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.61 - 1.50i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.38 - 2.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.95T + 31T^{2} \)
37 \( 1 + (-4.62 + 8.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.48 - 1.43i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.19 - 4.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + (-4.88 - 8.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 5.56T + 59T^{2} \)
61 \( 1 + 0.689iT - 61T^{2} \)
67 \( 1 - 4.11iT - 67T^{2} \)
71 \( 1 - 9.65iT - 71T^{2} \)
73 \( 1 + (-6.02 + 3.47i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 + (-2.52 - 4.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.06 + 3.50i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.316 + 0.182i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.22893460779625097864172698593, −9.454764308741738349389816938829, −8.722843287717471335602051404238, −7.76137309955930672302944500658, −7.29976203134220356548009772113, −6.24785993996747773021902046021, −5.00088512592986390386334421142, −3.85602501119689355865211844606, −3.12299878871451764785455805919, −2.24922561687841194098471368056, 0.21546180443889948772191649814, 2.07245639187039889009891601400, 3.22745499776324870785484047571, 4.17263200108024160138396200714, 4.82559722117386856199102635878, 6.52635269060528116744277725299, 7.19295595632605902732447200282, 7.927138084126058903474945034814, 8.674572844515691018246545918602, 9.627339640650323754691778406967

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