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Properties

Label 2-1008-63.59-c1-0-5
Degree $2$
Conductor $1008$
Sign $-0.998 - 0.0579i$
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  + (−1.42 + 0.979i)3-s − 1.68·5-s + (0.0236 + 2.64i)7-s + (1.07 − 2.79i)9-s + 3.90i·11-s + (5.24 + 3.02i)13-s + (2.40 − 1.65i)15-s + (0.201 − 0.348i)17-s + (0.145 − 0.0840i)19-s + (−2.62 − 3.75i)21-s − 8.88i·23-s − 2.15·25-s + (1.20 + 5.05i)27-s + (−6.15 + 3.55i)29-s + (−5.44 + 3.14i)31-s + ⋯
L(s)  = 1  + (−0.824 + 0.565i)3-s − 0.753·5-s + (0.00893 + 0.999i)7-s + (0.359 − 0.932i)9-s + 1.17i·11-s + (1.45 + 0.839i)13-s + (0.621 − 0.426i)15-s + (0.0488 − 0.0845i)17-s + (0.0334 − 0.0192i)19-s + (−0.573 − 0.819i)21-s − 1.85i·23-s − 0.431·25-s + (0.230 + 0.972i)27-s + (−1.14 + 0.659i)29-s + (−0.977 + 0.564i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.998 - 0.0579i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.998 - 0.0579i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5652089423\)
\(L(\frac12)\) \(\approx\) \(0.5652089423\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 - 0.979i)T \)
7 \( 1 + (-0.0236 - 2.64i)T \)
good5 \( 1 + 1.68T + 5T^{2} \)
11 \( 1 - 3.90iT - 11T^{2} \)
13 \( 1 + (-5.24 - 3.02i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.201 + 0.348i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.145 + 0.0840i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.88iT - 23T^{2} \)
29 \( 1 + (6.15 - 3.55i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.44 - 3.14i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.13 - 5.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.64 - 2.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.80 + 3.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.38 - 7.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.94 + 2.85i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.25 + 3.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.43 - 2.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.95 + 5.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (6.05 + 3.49i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.603 + 1.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.181 - 0.314i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.38 - 2.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.508 + 0.293i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.52811064393753664860663493647, −9.451310295103327156395161986129, −8.912049067108063740107079972813, −7.933971145006409346021588571465, −6.74874964506145147187914428095, −6.18219106356583217195672408014, −5.04573933569569743046936649314, −4.33410392989806717595174079263, −3.38197819372655763998022019765, −1.73630390548726996921065665275, 0.30737293248748368188663571883, 1.44821348520431663415422295394, 3.46777764874717296971229963663, 4.00927080948151244297024463244, 5.55790625738037541876951114992, 5.94646542567934965007326118301, 7.18317567101918776854967249155, 7.75674862387463665571734269663, 8.419312061231955958462395621667, 9.705462874689709702683477723733

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