Properties

Degree $2$
Conductor $1008$
Sign $-0.635 + 0.771i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.41i)3-s + (−1.72 + 2.98i)5-s + (−0.5 − 0.866i)7-s + (−1.00 + 2.82i)9-s + (1 + 1.73i)11-s + (2.44 − 4.24i)13-s + (5.94 − 0.548i)15-s + 2·17-s − 7.44·19-s + (−0.724 + 1.57i)21-s + (−0.5 + 0.866i)23-s + (−3.44 − 5.97i)25-s + (5.00 − 1.41i)27-s + (−1.44 − 2.51i)29-s + (3 − 5.19i)31-s + ⋯
L(s)  = 1  + (−0.577 − 0.816i)3-s + (−0.771 + 1.33i)5-s + (−0.188 − 0.327i)7-s + (−0.333 + 0.942i)9-s + (0.301 + 0.522i)11-s + (0.679 − 1.17i)13-s + (1.53 − 0.141i)15-s + 0.485·17-s − 1.70·19-s + (−0.158 + 0.343i)21-s + (−0.104 + 0.180i)23-s + (−0.689 − 1.19i)25-s + (0.962 − 0.272i)27-s + (−0.269 − 0.466i)29-s + (0.538 − 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.635 + 0.771i$
Motivic weight: \(1\)
Character: $\chi_{1008} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.635 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4837140750\)
\(L(\frac12)\) \(\approx\) \(0.4837140750\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.41i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.72 - 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.44 + 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 7.44T + 19T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.44 + 2.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.79T + 37T^{2} \)
41 \( 1 + (-4.89 + 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.44 + 2.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.89 + 8.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.10T + 53T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.72 + 9.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.55 - 2.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.89T + 71T^{2} \)
73 \( 1 - 2.89T + 73T^{2} \)
79 \( 1 + (-3.94 - 6.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.10T + 89T^{2} \)
97 \( 1 + (-3.44 - 5.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.05611126638474390580863235390, −8.498780186543142777612709893290, −7.80046011432453914023363357025, −7.08220885182221980249453396425, −6.44139223218738944001638154222, −5.62641154372072107758146933498, −4.19795613890689535319049717561, −3.25859243012756859283470059073, −2.06120870414923692069032507984, −0.25481528615209441411258965761, 1.32586673763996818821308854854, 3.35592605166668411807361210881, 4.31063365361766242026333753990, 4.78373234621885571599501331940, 5.95048934908979591808701958876, 6.60946371017500205569420558912, 8.091103648367152161693523441980, 8.878706649797077508464777449107, 9.116280954548618852841105159283, 10.31946807797550537313945752154

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