Properties

Label 2-2008-2008.331-c0-0-0
Degree $2$
Conductor $2008$
Sign $-0.132 + 0.991i$
Analytic cond. $1.00212$
Root an. cond. $1.00106$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.110 + 1.74i)3-s + (0.309 − 0.951i)4-s + (−1.11 − 1.35i)6-s + (0.309 + 0.951i)8-s + (−2.05 + 0.259i)9-s + (−1.80 − 0.462i)11-s + (1.69 + 0.435i)12-s + (−0.809 − 0.587i)16-s + (−1.27 + 1.54i)17-s + (1.51 − 1.41i)18-s + (0.688 − 1.46i)19-s + (1.72 − 0.684i)22-s + (−1.62 + 0.645i)24-s + (−0.809 + 0.587i)25-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.110 + 1.74i)3-s + (0.309 − 0.951i)4-s + (−1.11 − 1.35i)6-s + (0.309 + 0.951i)8-s + (−2.05 + 0.259i)9-s + (−1.80 − 0.462i)11-s + (1.69 + 0.435i)12-s + (−0.809 − 0.587i)16-s + (−1.27 + 1.54i)17-s + (1.51 − 1.41i)18-s + (0.688 − 1.46i)19-s + (1.72 − 0.684i)22-s + (−1.62 + 0.645i)24-s + (−0.809 + 0.587i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $-0.132 + 0.991i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ -0.132 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2046185429\)
\(L(\frac12)\) \(\approx\) \(0.2046185429\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
251 \( 1 + (0.187 - 0.982i)T \)
good3 \( 1 + (-0.110 - 1.74i)T + (-0.992 + 0.125i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.929 - 0.368i)T^{2} \)
11 \( 1 + (1.80 + 0.462i)T + (0.876 + 0.481i)T^{2} \)
13 \( 1 + (-0.968 + 0.248i)T^{2} \)
17 \( 1 + (1.27 - 1.54i)T + (-0.187 - 0.982i)T^{2} \)
19 \( 1 + (-0.688 + 1.46i)T + (-0.637 - 0.770i)T^{2} \)
23 \( 1 + (0.187 + 0.982i)T^{2} \)
29 \( 1 + (-0.0627 - 0.998i)T^{2} \)
31 \( 1 + (-0.876 - 0.481i)T^{2} \)
37 \( 1 + (-0.968 + 0.248i)T^{2} \)
41 \( 1 + (-0.348 - 0.137i)T + (0.728 + 0.684i)T^{2} \)
43 \( 1 + (1.11 + 0.614i)T + (0.535 + 0.844i)T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.535 + 0.844i)T^{2} \)
59 \( 1 + (1.23 + 1.49i)T + (-0.187 + 0.982i)T^{2} \)
61 \( 1 + (-0.535 + 0.844i)T^{2} \)
67 \( 1 + (-0.791 - 0.313i)T + (0.728 + 0.684i)T^{2} \)
71 \( 1 + (-0.968 + 0.248i)T^{2} \)
73 \( 1 + (0.101 - 1.61i)T + (-0.992 - 0.125i)T^{2} \)
79 \( 1 + (-0.0627 + 0.998i)T^{2} \)
83 \( 1 + (-0.0672 - 0.106i)T + (-0.425 + 0.904i)T^{2} \)
89 \( 1 + (0.929 - 1.12i)T + (-0.187 - 0.982i)T^{2} \)
97 \( 1 + (0.200 + 0.316i)T + (-0.425 + 0.904i)T^{2} \)
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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

−9.795990438850241049561269700640, −9.203717279065887827727199510999, −8.418801281058558803669138884466, −7.976874558029921450050238403053, −6.79193776001004992781966678755, −5.75956975131110932352983886108, −5.16667284824824943485588713440, −4.47644366548105229594995698108, −3.27718304299454088954917741462, −2.27821206917197101219200818097, 0.17238931423082641288108348808, 1.67508834049778948185072242269, 2.42952013723160627740892185845, 3.13119229863293148914572631714, 4.72571961844385336081784809368, 5.86227545459758213070292920977, 6.77890873794029807050874748086, 7.53623843397294162072406804479, 7.84524590679077389087992719967, 8.546556419878836789365592863452

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