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Properties

Label 2-2008-2008.1067-c0-0-0
Degree $2$
Conductor $2008$
Sign $-0.768 + 0.640i$
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 + (−0.824 + 0.211i)3-s + (0.309 + 0.951i)4-s + (0.791 + 0.313i)6-s + (0.309 − 0.951i)8-s + (−0.240 + 0.132i)9-s + (0.0672 + 0.106i)11-s + (−0.456 − 0.718i)12-s + (−0.809 + 0.587i)16-s + (−1.85 + 0.736i)17-s + (0.272 + 0.0344i)18-s + (0.303 − 1.58i)19-s + (0.00788 − 0.125i)22-s + (−0.0534 + 0.849i)24-s + (−0.809 − 0.587i)25-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.824 + 0.211i)3-s + (0.309 + 0.951i)4-s + (0.791 + 0.313i)6-s + (0.309 − 0.951i)8-s + (−0.240 + 0.132i)9-s + (0.0672 + 0.106i)11-s + (−0.456 − 0.718i)12-s + (−0.809 + 0.587i)16-s + (−1.85 + 0.736i)17-s + (0.272 + 0.0344i)18-s + (0.303 − 1.58i)19-s + (0.00788 − 0.125i)22-s + (−0.0534 + 0.849i)24-s + (−0.809 − 0.587i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $-0.768 + 0.640i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ -0.768 + 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2324823631\)
\(L(\frac12)\) \(\approx\) \(0.2324823631\)
\(L(1)\) 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 \( 1 + (-0.728 - 0.684i)T \)
good3 \( 1 + (0.824 - 0.211i)T + (0.876 - 0.481i)T^{2} \)
5 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.0627 + 0.998i)T^{2} \)
11 \( 1 + (-0.0672 - 0.106i)T + (-0.425 + 0.904i)T^{2} \)
13 \( 1 + (-0.535 + 0.844i)T^{2} \)
17 \( 1 + (1.85 - 0.736i)T + (0.728 - 0.684i)T^{2} \)
19 \( 1 + (-0.303 + 1.58i)T + (-0.929 - 0.368i)T^{2} \)
23 \( 1 + (-0.728 + 0.684i)T^{2} \)
29 \( 1 + (-0.968 + 0.248i)T^{2} \)
31 \( 1 + (0.425 - 0.904i)T^{2} \)
37 \( 1 + (-0.535 + 0.844i)T^{2} \)
41 \( 1 + (-0.0915 - 1.45i)T + (-0.992 + 0.125i)T^{2} \)
43 \( 1 + (-0.791 + 1.68i)T + (-0.637 - 0.770i)T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.637 - 0.770i)T^{2} \)
59 \( 1 + (0.996 + 0.394i)T + (0.728 + 0.684i)T^{2} \)
61 \( 1 + (0.637 - 0.770i)T^{2} \)
67 \( 1 + (0.0235 + 0.374i)T + (-0.992 + 0.125i)T^{2} \)
71 \( 1 + (-0.535 + 0.844i)T^{2} \)
73 \( 1 + (1.56 + 0.402i)T + (0.876 + 0.481i)T^{2} \)
79 \( 1 + (-0.968 - 0.248i)T^{2} \)
83 \( 1 + (1.23 + 1.49i)T + (-0.187 + 0.982i)T^{2} \)
89 \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \)
97 \( 1 + (0.929 + 1.12i)T + (-0.187 + 0.982i)T^{2} \)
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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

−8.962095478269382988348588313839, −8.587978076616104329897451536606, −7.54062657774354425391208541147, −6.70315580303097341567679703404, −6.05165780412953739721215612457, −4.80792420231093579258868268616, −4.17547197383157935846470513086, −2.85979693929909339141303643325, −1.95922980099398377633177722277, −0.24745098228046676060577755577, 1.33814891504065739918745781922, 2.59986439201017094853418072054, 4.12627397175734369669234295588, 5.18980942674268894189122472246, 5.91335580864846694419851498648, 6.45778267827388328555591270116, 7.29435443147518399074298220332, 8.004779164117706761499109343053, 8.970798784512960609298987835788, 9.411150273458656500917363698348

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