Properties

Label 2-2008-2008.339-c0-0-0
Degree $2$
Conductor $2008$
Sign $0.989 + 0.143i$
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  + (−0.992 − 0.125i)2-s + (0.216 − 0.527i)3-s + (0.968 + 0.248i)4-s + (−0.280 + 0.496i)6-s + (−0.929 − 0.368i)8-s + (0.480 + 0.474i)9-s + (0.891 + 0.302i)11-s + (0.340 − 0.456i)12-s + (0.876 + 0.481i)16-s + (−0.562 − 0.256i)17-s + (−0.417 − 0.530i)18-s + (0.635 + 0.567i)19-s + (−0.846 − 0.411i)22-s + (−0.395 + 0.410i)24-s + (−0.425 + 0.904i)25-s + ⋯
L(s)  = 1  + (−0.992 − 0.125i)2-s + (0.216 − 0.527i)3-s + (0.968 + 0.248i)4-s + (−0.280 + 0.496i)6-s + (−0.929 − 0.368i)8-s + (0.480 + 0.474i)9-s + (0.891 + 0.302i)11-s + (0.340 − 0.456i)12-s + (0.876 + 0.481i)16-s + (−0.562 − 0.256i)17-s + (−0.417 − 0.530i)18-s + (0.635 + 0.567i)19-s + (−0.846 − 0.411i)22-s + (−0.395 + 0.410i)24-s + (−0.425 + 0.904i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $0.989 + 0.143i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ 0.989 + 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8957651892\)
\(L(\frac12)\) \(\approx\) \(0.8957651892\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.992 + 0.125i)T \)
251 \( 1 + (0.514 + 0.857i)T \)
good3 \( 1 + (-0.216 + 0.527i)T + (-0.711 - 0.702i)T^{2} \)
5 \( 1 + (0.425 - 0.904i)T^{2} \)
7 \( 1 + (0.470 + 0.882i)T^{2} \)
11 \( 1 + (-0.891 - 0.302i)T + (0.793 + 0.607i)T^{2} \)
13 \( 1 + (-0.577 + 0.816i)T^{2} \)
17 \( 1 + (0.562 + 0.256i)T + (0.656 + 0.754i)T^{2} \)
19 \( 1 + (-0.635 - 0.567i)T + (0.112 + 0.993i)T^{2} \)
23 \( 1 + (-0.920 + 0.391i)T^{2} \)
29 \( 1 + (-0.762 - 0.647i)T^{2} \)
31 \( 1 + (0.999 + 0.0251i)T^{2} \)
37 \( 1 + (0.947 - 0.320i)T^{2} \)
41 \( 1 + (0.618 - 1.15i)T + (-0.556 - 0.830i)T^{2} \)
43 \( 1 + (-1.35 - 0.0339i)T + (0.998 + 0.0502i)T^{2} \)
47 \( 1 + (-0.535 + 0.844i)T^{2} \)
53 \( 1 + (0.778 - 0.627i)T^{2} \)
59 \( 1 + (0.0228 - 0.0104i)T + (0.656 - 0.754i)T^{2} \)
61 \( 1 + (0.837 + 0.546i)T^{2} \)
67 \( 1 + (-1.70 - 0.302i)T + (0.938 + 0.344i)T^{2} \)
71 \( 1 + (0.597 + 0.801i)T^{2} \)
73 \( 1 + (-0.0886 + 0.364i)T + (-0.888 - 0.459i)T^{2} \)
79 \( 1 + (-0.850 - 0.525i)T^{2} \)
83 \( 1 + (0.520 + 1.92i)T + (-0.863 + 0.503i)T^{2} \)
89 \( 1 + (0.00850 - 0.0748i)T + (-0.974 - 0.224i)T^{2} \)
97 \( 1 + (1.54 - 1.00i)T + (0.402 - 0.915i)T^{2} \)
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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

−9.416988540477123716385768420408, −8.489357211908860666205176932566, −7.82097313950160717924079105976, −7.11348041647228111367522898932, −6.57994779784434160625208936210, −5.53663222025366334622129696738, −4.29498215043917823771810075025, −3.22041992083334381507278666298, −2.07423507575622757258591333731, −1.29433988071119364317449777710, 1.05162899852608988171362773779, 2.35809418599449773469491740432, 3.49298061301716189997685377953, 4.32922360519393691396774731732, 5.54201506707945691714086910648, 6.50070118079256123014133170297, 6.98918875963823852856582067121, 7.984221836819412899624658986210, 8.787541616397058412230866464279, 9.328160187452284515591786680506

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