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Properties

Label 2-2008-2008.507-c0-0-0
Degree $2$
Conductor $2008$
Sign $0.289 - 0.957i$
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 + (−0.200 − 0.316i)3-s + (0.309 + 0.951i)4-s + (−0.0235 + 0.374i)6-s + (0.309 − 0.951i)8-s + (0.365 − 0.777i)9-s + (−1.23 + 1.49i)11-s + (0.238 − 0.288i)12-s + (−0.809 + 0.587i)16-s + (0.125 + 1.99i)17-s + (−0.753 + 0.414i)18-s + (−1.17 + 1.10i)19-s + (1.87 − 0.481i)22-s + (−0.362 + 0.0931i)24-s + (−0.809 − 0.587i)25-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.200 − 0.316i)3-s + (0.309 + 0.951i)4-s + (−0.0235 + 0.374i)6-s + (0.309 − 0.951i)8-s + (0.365 − 0.777i)9-s + (−1.23 + 1.49i)11-s + (0.238 − 0.288i)12-s + (−0.809 + 0.587i)16-s + (0.125 + 1.99i)17-s + (−0.753 + 0.414i)18-s + (−1.17 + 1.10i)19-s + (1.87 − 0.481i)22-s + (−0.362 + 0.0931i)24-s + (−0.809 − 0.587i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $0.289 - 0.957i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ 0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4013333926\)
\(L(\frac12)\) \(\approx\) \(0.4013333926\)
\(L(1)\) 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 \( 1 + (0.992 + 0.125i)T \)
good3 \( 1 + (0.200 + 0.316i)T + (-0.425 + 0.904i)T^{2} \)
5 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.968 + 0.248i)T^{2} \)
11 \( 1 + (1.23 - 1.49i)T + (-0.187 - 0.982i)T^{2} \)
13 \( 1 + (0.637 + 0.770i)T^{2} \)
17 \( 1 + (-0.125 - 1.99i)T + (-0.992 + 0.125i)T^{2} \)
19 \( 1 + (1.17 - 1.10i)T + (0.0627 - 0.998i)T^{2} \)
23 \( 1 + (0.992 - 0.125i)T^{2} \)
29 \( 1 + (-0.535 - 0.844i)T^{2} \)
31 \( 1 + (0.187 + 0.982i)T^{2} \)
37 \( 1 + (0.637 + 0.770i)T^{2} \)
41 \( 1 + (1.92 + 0.493i)T + (0.876 + 0.481i)T^{2} \)
43 \( 1 + (0.0235 + 0.123i)T + (-0.929 + 0.368i)T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.929 + 0.368i)T^{2} \)
59 \( 1 + (0.0800 - 1.27i)T + (-0.992 - 0.125i)T^{2} \)
61 \( 1 + (0.929 + 0.368i)T^{2} \)
67 \( 1 + (-1.41 - 0.362i)T + (0.876 + 0.481i)T^{2} \)
71 \( 1 + (0.637 + 0.770i)T^{2} \)
73 \( 1 + (0.866 - 1.36i)T + (-0.425 - 0.904i)T^{2} \)
79 \( 1 + (-0.535 + 0.844i)T^{2} \)
83 \( 1 + (0.996 - 0.394i)T + (0.728 - 0.684i)T^{2} \)
89 \( 1 + (-0.110 - 1.74i)T + (-0.992 + 0.125i)T^{2} \)
97 \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \)
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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

−9.778588003686723141570496034131, −8.581126934559874324687885935804, −8.126345722767450509118563553856, −7.29112212515513558430727363154, −6.57788606283027793058172649784, −5.69499523697950982079821916593, −4.27331772155923481773195743348, −3.73210470306256618105756686385, −2.26183222850281407941619805138, −1.62619200283082121889960387032, 0.36637089423054416663441595441, 2.14015869448479888366962193428, 3.10567865914948578738105420945, 4.76743582307981942110635986762, 5.18537284807430786774195143456, 6.03333144720534089012258526509, 7.01969362263656879944572491254, 7.68236105674877935362328372031, 8.395163379424047232464675990165, 9.087660572711462656384679428759

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