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Properties

Label 2-1008-63.47-c1-0-11
Degree $2$
Conductor $1008$
Sign $0.609 - 0.792i$
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  + (1.60 − 0.649i)3-s − 2.96·5-s + (−2.38 + 1.14i)7-s + (2.15 − 2.08i)9-s + 4.72i·11-s + (3.54 − 2.04i)13-s + (−4.76 + 1.92i)15-s + (0.835 + 1.44i)17-s + (4.25 + 2.45i)19-s + (−3.08 + 3.38i)21-s + 4.91i·23-s + 3.82·25-s + (2.11 − 4.74i)27-s + (0.238 + 0.137i)29-s + (1.38 + 0.801i)31-s + ⋯
L(s)  = 1  + (0.927 − 0.374i)3-s − 1.32·5-s + (−0.901 + 0.433i)7-s + (0.719 − 0.694i)9-s + 1.42i·11-s + (0.981 − 0.566i)13-s + (−1.23 + 0.497i)15-s + (0.202 + 0.350i)17-s + (0.975 + 0.563i)19-s + (−0.673 + 0.739i)21-s + 1.02i·23-s + 0.764·25-s + (0.406 − 0.913i)27-s + (0.0442 + 0.0255i)29-s + (0.249 + 0.143i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.609 - 0.792i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.609 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.489116542\)
\(L(\frac12)\) \(\approx\) \(1.489116542\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 + 0.649i)T \)
7 \( 1 + (2.38 - 1.14i)T \)
good5 \( 1 + 2.96T + 5T^{2} \)
11 \( 1 - 4.72iT - 11T^{2} \)
13 \( 1 + (-3.54 + 2.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.835 - 1.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.25 - 2.45i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.91iT - 23T^{2} \)
29 \( 1 + (-0.238 - 0.137i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.38 - 0.801i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.69 - 2.93i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.55 - 6.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.22 - 9.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.49 - 9.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.707 - 0.408i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.37 - 2.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.23 - 3.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.80 + 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (-13.6 + 7.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.15 + 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.03 - 6.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.60 - 7.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.00 + 4.04i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.749716388026786018733321773374, −9.354001851962463080321512843555, −8.133029590509830644354906156904, −7.79684218041512929979161761194, −6.94975091637102326467882995759, −5.99324226155515971360321221367, −4.54841164144400906765218304091, −3.57349291359759555975442296091, −3.00755986279058298644109298514, −1.40475320459284136256874821748, 0.67478958474280186144981243099, 2.77698685680093370193758397620, 3.68308616834701509241931079936, 4.01051550613142870779518063723, 5.43077298214494846092025843581, 6.74736650821052069887628653263, 7.37948740275380569156064483974, 8.454090125809580081632977149350, 8.738419283422652095429847759326, 9.752480210995603340597037758473

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