Properties

Degree $2$
Conductor $2008$
Sign $-0.137 - 0.990i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.5 + 1.53i)3-s + 4-s + (−0.5 + 1.53i)6-s + 8-s + (−1.30 − 0.951i)9-s + (0.190 − 0.587i)11-s + (−0.5 + 1.53i)12-s + 16-s + (0.618 + 1.90i)17-s + (−1.30 − 0.951i)18-s + (−1.61 + 1.17i)19-s + (0.190 − 0.587i)22-s + (−0.5 + 1.53i)24-s + 25-s + ⋯
L(s)  = 1  + 2-s + (−0.5 + 1.53i)3-s + 4-s + (−0.5 + 1.53i)6-s + 8-s + (−1.30 − 0.951i)9-s + (0.190 − 0.587i)11-s + (−0.5 + 1.53i)12-s + 16-s + (0.618 + 1.90i)17-s + (−1.30 − 0.951i)18-s + (−1.61 + 1.17i)19-s + (0.190 − 0.587i)22-s + (−0.5 + 1.53i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $-0.137 - 0.990i$
Motivic weight: \(0\)
Character: $\chi_{2008} (1619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ -0.137 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.855694768\)
\(L(\frac12)\) \(\approx\) \(1.855694768\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
251 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
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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

−9.964961833105443904181461007031, −8.687446837726954302103030513209, −8.208733157342159440285173067057, −6.80846697561595552973360330109, −5.98320338710540756900179282016, −5.55442959367847410153223555095, −4.58402567721183794783402779099, −3.82629587850914389270770732306, −3.43639917452265306101856813983, −1.89151238285185649856963161542, 1.10018684094431658516196510905, 2.28945656651594676027275666223, 2.96872576047878490064981224139, 4.53255540667671884101917998738, 5.08780076158096774088830567224, 6.13776501237703708797100137191, 6.75735513830182829714032355311, 7.20872443976209079613900885115, 7.933762312276523046538951497032, 8.981893290305750694354372118728

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