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Properties

Label 2-2008-2008.459-c0-0-0
Degree $2$
Conductor $2008$
Sign $-0.504 + 0.863i$
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  + (0.535 − 0.844i)2-s + (0.811 − 0.559i)3-s + (−0.425 − 0.904i)4-s + (−0.0371 − 0.984i)6-s + (−0.992 − 0.125i)8-s + (−0.0102 + 0.0269i)9-s + (1.96 − 0.198i)11-s + (−0.851 − 0.496i)12-s + (−0.637 + 0.770i)16-s + (−1.51 − 0.557i)17-s + (0.0172 + 0.0231i)18-s + (0.256 − 1.84i)19-s + (0.886 − 1.76i)22-s + (−0.875 + 0.452i)24-s + (−0.929 + 0.368i)25-s + ⋯
L(s)  = 1  + (0.535 − 0.844i)2-s + (0.811 − 0.559i)3-s + (−0.425 − 0.904i)4-s + (−0.0371 − 0.984i)6-s + (−0.992 − 0.125i)8-s + (−0.0102 + 0.0269i)9-s + (1.96 − 0.198i)11-s + (−0.851 − 0.496i)12-s + (−0.637 + 0.770i)16-s + (−1.51 − 0.557i)17-s + (0.0172 + 0.0231i)18-s + (0.256 − 1.84i)19-s + (0.886 − 1.76i)22-s + (−0.875 + 0.452i)24-s + (−0.929 + 0.368i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $-0.504 + 0.863i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ -0.504 + 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.917954171\)
\(L(\frac12)\) \(\approx\) \(1.917954171\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.535 + 0.844i)T \)
251 \( 1 + (0.997 - 0.0753i)T \)
good3 \( 1 + (-0.811 + 0.559i)T + (0.356 - 0.934i)T^{2} \)
5 \( 1 + (0.929 - 0.368i)T^{2} \)
7 \( 1 + (-0.988 - 0.150i)T^{2} \)
11 \( 1 + (-1.96 + 0.198i)T + (0.979 - 0.199i)T^{2} \)
13 \( 1 + (-0.402 + 0.915i)T^{2} \)
17 \( 1 + (1.51 + 0.557i)T + (0.762 + 0.647i)T^{2} \)
19 \( 1 + (-0.256 + 1.84i)T + (-0.962 - 0.272i)T^{2} \)
23 \( 1 + (0.236 + 0.971i)T^{2} \)
29 \( 1 + (0.999 + 0.0251i)T^{2} \)
31 \( 1 + (-0.112 + 0.993i)T^{2} \)
37 \( 1 + (-0.994 - 0.100i)T^{2} \)
41 \( 1 + (-1.50 + 0.229i)T + (0.954 - 0.297i)T^{2} \)
43 \( 1 + (0.125 - 1.10i)T + (-0.974 - 0.224i)T^{2} \)
47 \( 1 + (0.187 - 0.982i)T^{2} \)
53 \( 1 + (0.0878 - 0.996i)T^{2} \)
59 \( 1 + (1.40 - 0.514i)T + (0.762 - 0.647i)T^{2} \)
61 \( 1 + (0.514 + 0.857i)T^{2} \)
67 \( 1 + (0.987 - 0.974i)T + (0.0125 - 0.999i)T^{2} \)
71 \( 1 + (0.863 - 0.503i)T^{2} \)
73 \( 1 + (0.0357 - 0.120i)T + (-0.837 - 0.546i)T^{2} \)
79 \( 1 + (-0.793 + 0.607i)T^{2} \)
83 \( 1 + (0.612 - 0.260i)T + (0.693 - 0.720i)T^{2} \)
89 \( 1 + (1.11 - 0.315i)T + (0.850 - 0.525i)T^{2} \)
97 \( 1 + (-0.243 + 0.405i)T + (-0.470 - 0.882i)T^{2} \)
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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

−9.055261468310540406576843388410, −8.754215123301578703360866886338, −7.39748632147050668297681309234, −6.73322889896908728939680841580, −5.92280201933655457746199226061, −4.68944548935894394758936299370, −4.08347505034962669472635624119, −2.98308018809828308360931304622, −2.26447068947403157661095638350, −1.19858676798303225309400283493, 1.94281143273989920136115306411, 3.31819048962052333066477262421, 4.08435854177126656706698617951, 4.32787239161312108205414981918, 5.91399468367098400348087281598, 6.28391804065815535590897079153, 7.21815243425318975938169174091, 8.134696805156976769819426955112, 8.810054598088197624708174270742, 9.290983981243171840548399555866

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