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Properties

Label 2-1008-252.139-c1-0-43
Degree $2$
Conductor $1008$
Sign $-0.691 + 0.722i$
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  + (1.11 − 1.32i)3-s + (0.263 + 0.152i)5-s + (−2.36 + 1.17i)7-s + (−0.510 − 2.95i)9-s + (−0.437 + 0.252i)11-s + (−4.11 − 2.37i)13-s + (0.495 − 0.179i)15-s − 4.53i·17-s + 4.38·19-s + (−1.07 + 4.45i)21-s + (−2.76 − 1.59i)23-s + (−2.45 − 4.24i)25-s + (−4.48 − 2.62i)27-s + (−4.86 − 8.43i)29-s + (1.83 − 3.18i)31-s + ⋯
L(s)  = 1  + (0.644 − 0.764i)3-s + (0.117 + 0.0680i)5-s + (−0.895 + 0.445i)7-s + (−0.170 − 0.985i)9-s + (−0.131 + 0.0761i)11-s + (−1.13 − 0.658i)13-s + (0.127 − 0.0462i)15-s − 1.09i·17-s + 1.00·19-s + (−0.235 + 0.971i)21-s + (−0.577 − 0.333i)23-s + (−0.490 − 0.849i)25-s + (−0.863 − 0.504i)27-s + (−0.904 − 1.56i)29-s + (0.330 − 0.572i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.218609298\)
\(L(\frac12)\) \(\approx\) \(1.218609298\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 1.32i)T \)
7 \( 1 + (2.36 - 1.17i)T \)
good5 \( 1 + (-0.263 - 0.152i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.437 - 0.252i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.11 + 2.37i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.53iT - 17T^{2} \)
19 \( 1 - 4.38T + 19T^{2} \)
23 \( 1 + (2.76 + 1.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.86 + 8.43i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.83 + 3.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.70T + 37T^{2} \)
41 \( 1 + (-7.01 - 4.05i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.99 + 2.30i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.23 - 5.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.52T + 53T^{2} \)
59 \( 1 + (6.40 - 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.16 - 1.24i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.03 + 3.48i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + 7.13iT - 73T^{2} \)
79 \( 1 + (5.96 - 3.44i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.66 - 6.35i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.8iT - 89T^{2} \)
97 \( 1 + (-10.9 + 6.34i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.647201384559685261938303863301, −8.920784395626717923355375694419, −7.68894492049824064830020486167, −7.43371985955512668764982815972, −6.24880667799186689768904918991, −5.58839402618350693030058763636, −4.17851933075502886844370883058, −2.85781233206167191001328746055, −2.40105489243642293620275302985, −0.48048807576772511590371763130, 1.90191753983214935402151147369, 3.21002401756725825554503737464, 3.87912732244632706766092711859, 4.97557802309798586047625201921, 5.86975252525035510192289687407, 7.15474983648119229003173048141, 7.67802310933250028871125046955, 8.957756636102007496187906922415, 9.380364840701491231645898731574, 10.18912999547121503133895741535

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