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Properties

Label 2-1008-48.11-c1-0-17
Degree $2$
Conductor $1008$
Sign $-0.662 - 0.748i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.662 - 0.748i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.662 - 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.209492230\)
\(L(\frac12)\) \(\approx\) \(2.209492230\)
\(L(\frac{3}{2})\) 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

−10.40215231624537268417180842200, −9.242067391615598229270484321045, −8.401388323208750668577638535775, −7.50479838006857687304220070509, −6.96033501998846272017922302299, −5.98215617716372109166986846794, −5.07545719628716882842495752458, −4.08295236045917826064263943262, −3.34128228372619455038980542328, −1.95199691026546034938916122003, 0.791819538675850200700754269192, 2.17977867625191574969295642848, 3.40739374681777429411248085174, 4.34396960283839157065714526446, 5.03407326000732222152126501507, 6.13500889671606955306399275842, 6.86776090304979588838031238850, 8.161353364598208432301299847170, 8.843125172622037303488333550101, 9.859380153128642780312415404264

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