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Properties

Label 2-2008-2008.443-c0-0-0
Degree $2$
Conductor $2008$
Sign $-0.943 - 0.332i$
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  + (−0.187 − 0.982i)2-s + (−0.834 − 0.911i)3-s + (−0.929 + 0.368i)4-s + (−0.738 + 0.990i)6-s + (0.535 + 0.844i)8-s + (−0.0462 + 0.524i)9-s + (0.230 − 1.65i)11-s + (1.11 + 0.540i)12-s + (0.728 − 0.684i)16-s + (0.590 + 0.183i)17-s + (0.524 − 0.0528i)18-s + (1.41 − 1.39i)19-s + (−1.67 + 0.0841i)22-s + (0.322 − 1.19i)24-s + (−0.992 − 0.125i)25-s + ⋯
L(s)  = 1  + (−0.187 − 0.982i)2-s + (−0.834 − 0.911i)3-s + (−0.929 + 0.368i)4-s + (−0.738 + 0.990i)6-s + (0.535 + 0.844i)8-s + (−0.0462 + 0.524i)9-s + (0.230 − 1.65i)11-s + (1.11 + 0.540i)12-s + (0.728 − 0.684i)16-s + (0.590 + 0.183i)17-s + (0.524 − 0.0528i)18-s + (1.41 − 1.39i)19-s + (−1.67 + 0.0841i)22-s + (0.322 − 1.19i)24-s + (−0.992 − 0.125i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $-0.943 - 0.332i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ -0.943 - 0.332i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6219213426\)
\(L(\frac12)\) \(\approx\) \(0.6219213426\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.187 + 0.982i)T \)
251 \( 1 + (0.285 + 0.958i)T \)
good3 \( 1 + (0.834 + 0.911i)T + (-0.0878 + 0.996i)T^{2} \)
5 \( 1 + (0.992 + 0.125i)T^{2} \)
7 \( 1 + (0.837 + 0.546i)T^{2} \)
11 \( 1 + (-0.230 + 1.65i)T + (-0.962 - 0.272i)T^{2} \)
13 \( 1 + (-0.693 - 0.720i)T^{2} \)
17 \( 1 + (-0.590 - 0.183i)T + (0.823 + 0.567i)T^{2} \)
19 \( 1 + (-1.41 + 1.39i)T + (0.0125 - 0.999i)T^{2} \)
23 \( 1 + (-0.793 + 0.607i)T^{2} \)
29 \( 1 + (0.910 - 0.414i)T^{2} \)
31 \( 1 + (-0.938 + 0.344i)T^{2} \)
37 \( 1 + (0.137 + 0.990i)T^{2} \)
41 \( 1 + (1.37 - 0.899i)T + (0.402 - 0.915i)T^{2} \)
43 \( 1 + (-1.08 + 0.398i)T + (0.762 - 0.647i)T^{2} \)
47 \( 1 + (-0.0627 - 0.998i)T^{2} \)
53 \( 1 + (0.997 + 0.0753i)T^{2} \)
59 \( 1 + (1.88 - 0.584i)T + (0.823 - 0.567i)T^{2} \)
61 \( 1 + (0.236 + 0.971i)T^{2} \)
67 \( 1 + (0.698 + 0.562i)T + (0.212 + 0.977i)T^{2} \)
71 \( 1 + (-0.899 + 0.437i)T^{2} \)
73 \( 1 + (-1.71 + 0.350i)T + (0.920 - 0.391i)T^{2} \)
79 \( 1 + (-0.112 + 0.993i)T^{2} \)
83 \( 1 + (-0.838 - 0.517i)T + (0.448 + 0.893i)T^{2} \)
89 \( 1 + (0.0217 + 1.72i)T + (-0.999 + 0.0251i)T^{2} \)
97 \( 1 + (0.375 - 1.54i)T + (-0.888 - 0.459i)T^{2} \)
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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

−9.071330579363648723346552074853, −8.128649897672571194989697943594, −7.49075215155822035182845403318, −6.44937401878281451016477971741, −5.68529678310568237667158682119, −4.97166665774837032173788851482, −3.63252543754838256331791969719, −2.93219773778267566390389748835, −1.51511313136227737313114662905, −0.59078457030916367090613237170, 1.61505874337084722630624122255, 3.58859765004635090871969478808, 4.36294899023436486184593444093, 5.13940222827165872994680308181, 5.66322943955396548179701932748, 6.54596335800253259062704573228, 7.55547425176874907783815266827, 7.890839341094889049001274330587, 9.272852933063793727746334766469, 9.778665358923640193907905430329

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