Properties

Degree $2$
Conductor $2008$
Sign $0.913 + 0.405i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.5 − 0.363i)3-s + 4-s + (−0.5 − 0.363i)6-s + 8-s + (−0.190 − 0.587i)9-s + (1.30 + 0.951i)11-s + (−0.5 − 0.363i)12-s + 16-s + (−1.61 + 1.17i)17-s + (−0.190 − 0.587i)18-s + (0.618 − 1.90i)19-s + (1.30 + 0.951i)22-s + (−0.5 − 0.363i)24-s + 25-s + ⋯
L(s)  = 1  + 2-s + (−0.5 − 0.363i)3-s + 4-s + (−0.5 − 0.363i)6-s + 8-s + (−0.190 − 0.587i)9-s + (1.30 + 0.951i)11-s + (−0.5 − 0.363i)12-s + 16-s + (−1.61 + 1.17i)17-s + (−0.190 − 0.587i)18-s + (0.618 − 1.90i)19-s + (1.30 + 0.951i)22-s + (−0.5 − 0.363i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $0.913 + 0.405i$
Motivic weight: \(0\)
Character: $\chi_{2008} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ 0.913 + 0.405i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.970744002\)
\(L(\frac12)\) \(\approx\) \(1.970744002\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
251 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
11 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
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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

−9.173643097047870049428652811111, −8.624285374357432262692624621824, −7.07655257765350887553915933671, −6.86868107128889817792849155034, −6.27357892238035025396747281397, −5.20229993260623766015317565465, −4.43798917687498245782455892063, −3.66792740311743405604467278088, −2.47263942200768363886577483638, −1.36425799060875297604548973382, 1.53126557955658848634939012762, 2.82380847350095772688315132651, 3.74758887455869523144583677496, 4.59661846145463986290395199498, 5.28747670265424239803874419285, 6.17586723183085153684699797146, 6.66366075442673146063393648845, 7.70659402097243230279899556184, 8.552272599108721182956403896577, 9.504493176688873358187124725386

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