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Properties

Label 2-1008-9.7-c1-0-17
Degree $2$
Conductor $1008$
Sign $0.998 - 0.0529i$
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 − 0.255i)3-s + (1.81 + 3.13i)5-s + (0.5 − 0.866i)7-s + (2.86 + 0.875i)9-s + (1.95 − 3.39i)11-s + (−2.53 − 4.39i)13-s + (−2.30 − 5.83i)15-s + 1.03·17-s + 2.50·19-s + (−1.07 + 1.35i)21-s + (2.47 + 4.29i)23-s + (−4.06 + 7.04i)25-s + (−4.69 − 2.23i)27-s + (4.60 − 7.97i)29-s + (−0.422 − 0.731i)31-s + ⋯
L(s)  = 1  + (−0.989 − 0.147i)3-s + (0.810 + 1.40i)5-s + (0.188 − 0.327i)7-s + (0.956 + 0.291i)9-s + (0.590 − 1.02i)11-s + (−0.703 − 1.21i)13-s + (−0.594 − 1.50i)15-s + 0.250·17-s + 0.575·19-s + (−0.235 + 0.295i)21-s + (0.516 + 0.895i)23-s + (−0.813 + 1.40i)25-s + (−0.902 − 0.429i)27-s + (0.854 − 1.48i)29-s + (−0.0758 − 0.131i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.998 - 0.0529i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.998 - 0.0529i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.430235451\)
\(L(\frac12)\) \(\approx\) \(1.430235451\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.255i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.81 - 3.13i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.95 + 3.39i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.53 + 4.39i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.03T + 17T^{2} \)
19 \( 1 - 2.50T + 19T^{2} \)
23 \( 1 + (-2.47 - 4.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.60 + 7.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.422 + 0.731i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.84T + 37T^{2} \)
41 \( 1 + (-2.07 - 3.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.20 - 3.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.93 - 6.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + (5.60 + 9.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.208 - 0.360i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.02 - 8.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.05T + 71T^{2} \)
73 \( 1 - 7.20T + 73T^{2} \)
79 \( 1 + (-7.56 + 13.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.932 + 1.61i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.669T + 89T^{2} \)
97 \( 1 + (7.63 - 13.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.974463188783365509474824841174, −9.658260321200064995863145891672, −7.992193523011680517008559261638, −7.33605400977824105954321826829, −6.36977546219676106848934342645, −5.90887080692705977292652798543, −4.99341134149126926130745149952, −3.56041807731883871826669307090, −2.57238193044226460596354933740, −0.977220995333953795482071952560, 1.11070930205010049140711775876, 2.08586393242414142155157691636, 4.16844378365029645867305905813, 4.91749343545374917000943027908, 5.36412096112139987816640096344, 6.54105891187425255837441999525, 7.16039884782235399277522929786, 8.578337201984243619366910815328, 9.323513544162589419084711708276, 9.767109919040719141391788275727

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