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Properties

Label 2-2008-2008.627-c0-0-0
Degree $2$
Conductor $2008$
Sign $-0.744 + 0.667i$
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  + (0.309 − 0.951i)2-s + (−1.06 + 0.134i)3-s + (−0.809 − 0.587i)4-s + (−0.200 + 1.05i)6-s + (−0.809 + 0.587i)8-s + (0.143 − 0.0369i)9-s + (1.27 + 0.702i)11-s + (0.939 + 0.516i)12-s + (0.309 + 0.951i)16-s + (−0.374 − 1.96i)17-s + (0.00932 − 0.148i)18-s + (−0.393 − 0.476i)19-s + (1.06 − 0.998i)22-s + (0.781 − 0.733i)24-s + (0.309 − 0.951i)25-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−1.06 + 0.134i)3-s + (−0.809 − 0.587i)4-s + (−0.200 + 1.05i)6-s + (−0.809 + 0.587i)8-s + (0.143 − 0.0369i)9-s + (1.27 + 0.702i)11-s + (0.939 + 0.516i)12-s + (0.309 + 0.951i)16-s + (−0.374 − 1.96i)17-s + (0.00932 − 0.148i)18-s + (−0.393 − 0.476i)19-s + (1.06 − 0.998i)22-s + (0.781 − 0.733i)24-s + (0.309 − 0.951i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $-0.744 + 0.667i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ -0.744 + 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7082290632\)
\(L(\frac12)\) \(\approx\) \(0.7082290632\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
251 \( 1 + (0.929 + 0.368i)T \)
good3 \( 1 + (1.06 - 0.134i)T + (0.968 - 0.248i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.728 + 0.684i)T^{2} \)
11 \( 1 + (-1.27 - 0.702i)T + (0.535 + 0.844i)T^{2} \)
13 \( 1 + (-0.876 + 0.481i)T^{2} \)
17 \( 1 + (0.374 + 1.96i)T + (-0.929 + 0.368i)T^{2} \)
19 \( 1 + (0.393 + 0.476i)T + (-0.187 + 0.982i)T^{2} \)
23 \( 1 + (0.929 - 0.368i)T^{2} \)
29 \( 1 + (0.992 - 0.125i)T^{2} \)
31 \( 1 + (-0.535 - 0.844i)T^{2} \)
37 \( 1 + (-0.876 + 0.481i)T^{2} \)
41 \( 1 + (1.35 + 1.27i)T + (0.0627 + 0.998i)T^{2} \)
43 \( 1 + (0.200 + 0.316i)T + (-0.425 + 0.904i)T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.425 + 0.904i)T^{2} \)
59 \( 1 + (0.328 - 1.72i)T + (-0.929 - 0.368i)T^{2} \)
61 \( 1 + (0.425 + 0.904i)T^{2} \)
67 \( 1 + (0.929 + 0.872i)T + (0.0627 + 0.998i)T^{2} \)
71 \( 1 + (-0.876 + 0.481i)T^{2} \)
73 \( 1 + (0.613 + 0.0774i)T + (0.968 + 0.248i)T^{2} \)
79 \( 1 + (0.992 + 0.125i)T^{2} \)
83 \( 1 + (-0.844 + 1.79i)T + (-0.637 - 0.770i)T^{2} \)
89 \( 1 + (0.0235 + 0.123i)T + (-0.929 + 0.368i)T^{2} \)
97 \( 1 + (-0.791 + 1.68i)T + (-0.637 - 0.770i)T^{2} \)
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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

−9.129208662412940333914474172156, −8.734476354765283161134491871917, −7.20008187456567244798568590972, −6.54409551975560669555894992308, −5.65180860314692012360470313800, −4.78306617997606555327675925369, −4.35879187252543665461696424682, −3.10431851756514473414143883075, −2.02894913013067076729499434238, −0.58311975489442036627797167741, 1.36200027358655626369629912297, 3.35148297252947183584456890037, 4.08267514717864797739990687076, 5.04267564913613653484584458982, 5.97476906130045945513205236376, 6.28861360892564090864483069075, 6.93045540889147108312874829482, 8.125248008007057230657189039608, 8.629345461521974538221606000318, 9.434974515754979599667353303430

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