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Properties

Label 2-1008-48.35-c1-0-11
Degree $2$
Conductor $1008$
Sign $-0.484 - 0.874i$
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 + 0.866i)2-s + (0.496 − 1.93i)4-s + (0.925 + 0.925i)5-s + 7-s + (1.12 + 2.59i)8-s + (−1.83 − 0.231i)10-s + (−1.72 + 1.72i)11-s + (0.328 + 0.328i)13-s + (−1.11 + 0.866i)14-s + (−3.50 − 1.92i)16-s + 2.34i·17-s + (−1.77 + 1.77i)19-s + (2.25 − 1.33i)20-s + (0.431 − 3.42i)22-s + 6.17i·23-s + ⋯
L(s)  = 1  + (−0.790 + 0.613i)2-s + (0.248 − 0.968i)4-s + (0.413 + 0.413i)5-s + 0.377·7-s + (0.397 + 0.917i)8-s + (−0.580 − 0.0732i)10-s + (−0.519 + 0.519i)11-s + (0.0910 + 0.0910i)13-s + (−0.298 + 0.231i)14-s + (−0.876 − 0.481i)16-s + 0.567i·17-s + (−0.408 + 0.408i)19-s + (0.503 − 0.298i)20-s + (0.0920 − 0.729i)22-s + 1.28i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.484 - 0.874i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.484 - 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9813279082\)
\(L(\frac12)\) \(\approx\) \(0.9813279082\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 - 0.866i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (-0.925 - 0.925i)T + 5iT^{2} \)
11 \( 1 + (1.72 - 1.72i)T - 11iT^{2} \)
13 \( 1 + (-0.328 - 0.328i)T + 13iT^{2} \)
17 \( 1 - 2.34iT - 17T^{2} \)
19 \( 1 + (1.77 - 1.77i)T - 19iT^{2} \)
23 \( 1 - 6.17iT - 23T^{2} \)
29 \( 1 + (0.122 - 0.122i)T - 29iT^{2} \)
31 \( 1 - 1.74iT - 31T^{2} \)
37 \( 1 + (1.68 - 1.68i)T - 37iT^{2} \)
41 \( 1 - 2.88T + 41T^{2} \)
43 \( 1 + (-2.77 - 2.77i)T + 43iT^{2} \)
47 \( 1 - 5.92T + 47T^{2} \)
53 \( 1 + (0.973 + 0.973i)T + 53iT^{2} \)
59 \( 1 + (8.33 - 8.33i)T - 59iT^{2} \)
61 \( 1 + (-4.28 - 4.28i)T + 61iT^{2} \)
67 \( 1 + (-1.78 + 1.78i)T - 67iT^{2} \)
71 \( 1 - 8.57iT - 71T^{2} \)
73 \( 1 - 6.41iT - 73T^{2} \)
79 \( 1 - 5.38iT - 79T^{2} \)
83 \( 1 + (3.46 + 3.46i)T + 83iT^{2} \)
89 \( 1 + 1.51T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−10.18658026703153678672247825732, −9.411826990770720287004952465904, −8.500938398913285672483208772001, −7.75359700650830395804037085675, −7.00217123298436270326453993737, −6.06934810312922266953256161598, −5.36344352659258516242080191694, −4.21498837978008659273819010052, −2.56169855995675554366514275335, −1.49103427458192205789102453506, 0.59417842478559522339769354995, 2.00424101805755211644631063306, 2.99526972332116903035987840375, 4.26630720537816715604582362621, 5.27298564009275931173733443904, 6.41802054396254810763899103740, 7.44094319949178520083936129692, 8.236538911382548770283012303780, 8.964617007190674413246192911717, 9.585343294082858902506494812111

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