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Properties

Label 2-2008-2008.67-c0-0-0
Degree $2$
Conductor $2008$
Sign $0.760 + 0.648i$
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  + (0.728 − 0.684i)2-s + (−0.422 + 0.791i)3-s + (0.0627 − 0.998i)4-s + (0.233 + 0.865i)6-s + (−0.637 − 0.770i)8-s + (0.109 + 0.162i)9-s + (0.751 − 0.637i)11-s + (0.763 + 0.470i)12-s + (−0.992 − 0.125i)16-s + (−0.481 + 0.387i)17-s + (0.190 + 0.0439i)18-s + (1.44 − 0.994i)19-s + (0.111 − 0.979i)22-s + (0.878 − 0.179i)24-s + (0.876 + 0.481i)25-s + ⋯
L(s)  = 1  + (0.728 − 0.684i)2-s + (−0.422 + 0.791i)3-s + (0.0627 − 0.998i)4-s + (0.233 + 0.865i)6-s + (−0.637 − 0.770i)8-s + (0.109 + 0.162i)9-s + (0.751 − 0.637i)11-s + (0.763 + 0.470i)12-s + (−0.992 − 0.125i)16-s + (−0.481 + 0.387i)17-s + (0.190 + 0.0439i)18-s + (1.44 − 0.994i)19-s + (0.111 − 0.979i)22-s + (0.878 − 0.179i)24-s + (0.876 + 0.481i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $0.760 + 0.648i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ 0.760 + 0.648i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.574348930\)
\(L(\frac12)\) \(\approx\) \(1.574348930\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.728 + 0.684i)T \)
251 \( 1 + (0.863 - 0.503i)T \)
good3 \( 1 + (0.422 - 0.791i)T + (-0.556 - 0.830i)T^{2} \)
5 \( 1 + (-0.876 - 0.481i)T^{2} \)
7 \( 1 + (-0.492 - 0.870i)T^{2} \)
11 \( 1 + (-0.751 + 0.637i)T + (0.162 - 0.986i)T^{2} \)
13 \( 1 + (0.236 + 0.971i)T^{2} \)
17 \( 1 + (0.481 - 0.387i)T + (0.212 - 0.977i)T^{2} \)
19 \( 1 + (-1.44 + 0.994i)T + (0.356 - 0.934i)T^{2} \)
23 \( 1 + (-0.994 + 0.100i)T^{2} \)
29 \( 1 + (0.984 + 0.175i)T^{2} \)
31 \( 1 + (0.711 - 0.702i)T^{2} \)
37 \( 1 + (-0.762 - 0.647i)T^{2} \)
41 \( 1 + (-0.208 + 0.368i)T + (-0.514 - 0.857i)T^{2} \)
43 \( 1 + (-1.19 + 1.17i)T + (0.0125 - 0.999i)T^{2} \)
47 \( 1 + (-0.968 + 0.248i)T^{2} \)
53 \( 1 + (-0.577 + 0.816i)T^{2} \)
59 \( 1 + (-0.591 - 0.476i)T + (0.212 + 0.977i)T^{2} \)
61 \( 1 + (0.597 + 0.801i)T^{2} \)
67 \( 1 + (1.07 + 1.17i)T + (-0.0878 + 0.996i)T^{2} \)
71 \( 1 + (-0.850 + 0.525i)T^{2} \)
73 \( 1 + (0.765 + 0.372i)T + (0.617 + 0.786i)T^{2} \)
79 \( 1 + (0.137 - 0.990i)T^{2} \)
83 \( 1 + (1.31 + 0.445i)T + (0.793 + 0.607i)T^{2} \)
89 \( 1 + (-0.655 - 1.71i)T + (-0.745 + 0.666i)T^{2} \)
97 \( 1 + (1.18 - 1.59i)T + (-0.285 - 0.958i)T^{2} \)
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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

−9.185775069872134952631759953554, −9.086974818758693712597517692144, −7.53096784245294594055725288236, −6.67847877186565179658013625086, −5.72807285709328232443146995776, −5.12817785917448147329649138781, −4.30796534698551858853971581541, −3.58945848521309382230597967652, −2.58754898802311294665089099687, −1.17640774150379464498103464986, 1.38129229101245364081358755086, 2.74037875898951784325262118617, 3.84865164914893556539873042498, 4.63932781545415301800981358331, 5.65453815835419918832605207172, 6.27682641426479876945411797676, 7.11793609069695698988112258095, 7.40970894721127104822991123939, 8.432403316620593267025652104173, 9.295768674379211371942322412686

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