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Properties

Label 2-2008-2008.403-c0-0-0
Degree $2$
Conductor $2008$
Sign $0.182 - 0.983i$
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  + (0.968 − 0.248i)2-s + (−0.253 + 1.53i)3-s + (0.876 − 0.481i)4-s + (0.136 + 1.55i)6-s + (0.728 − 0.684i)8-s + (−1.34 − 0.457i)9-s + (−0.535 + 1.79i)11-s + (0.518 + 1.46i)12-s + (0.535 − 0.844i)16-s + (1.57 − 0.362i)17-s + (−1.41 − 0.107i)18-s + (−1.24 + 0.254i)19-s + (−0.0707 + 1.87i)22-s + (0.867 + 1.29i)24-s + (−0.637 + 0.770i)25-s + ⋯
L(s)  = 1  + (0.968 − 0.248i)2-s + (−0.253 + 1.53i)3-s + (0.876 − 0.481i)4-s + (0.136 + 1.55i)6-s + (0.728 − 0.684i)8-s + (−1.34 − 0.457i)9-s + (−0.535 + 1.79i)11-s + (0.518 + 1.46i)12-s + (0.535 − 0.844i)16-s + (1.57 − 0.362i)17-s + (−1.41 − 0.107i)18-s + (−1.24 + 0.254i)19-s + (−0.0707 + 1.87i)22-s + (0.867 + 1.29i)24-s + (−0.637 + 0.770i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $0.182 - 0.983i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ 0.182 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.898503484\)
\(L(\frac12)\) \(\approx\) \(1.898503484\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.968 + 0.248i)T \)
251 \( 1 + (0.984 - 0.175i)T \)
good3 \( 1 + (0.253 - 1.53i)T + (-0.947 - 0.320i)T^{2} \)
5 \( 1 + (0.637 - 0.770i)T^{2} \)
7 \( 1 + (-0.938 - 0.344i)T^{2} \)
11 \( 1 + (0.535 - 1.79i)T + (-0.837 - 0.546i)T^{2} \)
13 \( 1 + (-0.823 + 0.567i)T^{2} \)
17 \( 1 + (-1.57 + 0.362i)T + (0.899 - 0.437i)T^{2} \)
19 \( 1 + (1.24 - 0.254i)T + (0.920 - 0.391i)T^{2} \)
23 \( 1 + (0.470 - 0.882i)T^{2} \)
29 \( 1 + (-0.448 - 0.893i)T^{2} \)
31 \( 1 + (-0.260 + 0.965i)T^{2} \)
37 \( 1 + (0.285 + 0.958i)T^{2} \)
41 \( 1 + (-1.68 + 0.619i)T + (0.762 - 0.647i)T^{2} \)
43 \( 1 + (-0.342 + 1.26i)T + (-0.863 - 0.503i)T^{2} \)
47 \( 1 + (0.425 - 0.904i)T^{2} \)
53 \( 1 + (0.745 + 0.666i)T^{2} \)
59 \( 1 + (1.54 + 0.356i)T + (0.899 + 0.437i)T^{2} \)
61 \( 1 + (-0.212 - 0.977i)T^{2} \)
67 \( 1 + (-1.29 - 0.368i)T + (0.850 + 0.525i)T^{2} \)
71 \( 1 + (0.332 - 0.942i)T^{2} \)
73 \( 1 + (1.83 + 0.279i)T + (0.954 + 0.297i)T^{2} \)
79 \( 1 + (0.888 - 0.459i)T^{2} \)
83 \( 1 + (0.573 + 1.30i)T + (-0.675 + 0.737i)T^{2} \)
89 \( 1 + (0.698 + 0.297i)T + (0.693 + 0.720i)T^{2} \)
97 \( 1 + (0.199 - 0.920i)T + (-0.910 - 0.414i)T^{2} \)
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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

−9.811710762868436976769219784920, −9.142081610903819461127565656505, −7.73507026723206908290317162864, −7.13604282365234875991027884857, −5.83400371479784521886231431787, −5.37477973201503792413725786508, −4.46550966656306153428401087595, −4.06721609652306488696854795339, −3.04574017687460854066818102568, −1.94481133327273403495045358252, 1.09314756218884787403767818880, 2.38716034285016077195646365371, 3.15648864140953397354469128135, 4.28228701473109990794802054307, 5.63931426162123945794799107512, 5.95063276439509596444953305769, 6.56460044859419511288710456441, 7.66133573918623345017548540995, 7.968549402583672719452243104897, 8.691583665445313092659274302423

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