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Properties

Label 2-2004-2004.2003-c0-0-1
Degree $2$
Conductor $2004$
Sign $-0.654 + 0.755i$
Analytic cond. $1.00012$
Root an. cond. $1.00006$
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  + (−0.415 + 0.909i)2-s + (0.142 + 0.989i)3-s + (−0.654 − 0.755i)4-s + (−0.959 − 0.281i)6-s + 1.08i·7-s + (0.959 − 0.281i)8-s + (−0.959 + 0.281i)9-s − 1.30·11-s + (0.654 − 0.755i)12-s + (−0.983 − 0.449i)14-s + (−0.142 + 0.989i)16-s + (0.142 − 0.989i)18-s + 0.563i·19-s + (−1.07 + 0.153i)21-s + (0.544 − 1.19i)22-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (0.142 + 0.989i)3-s + (−0.654 − 0.755i)4-s + (−0.959 − 0.281i)6-s + 1.08i·7-s + (0.959 − 0.281i)8-s + (−0.959 + 0.281i)9-s − 1.30·11-s + (0.654 − 0.755i)12-s + (−0.983 − 0.449i)14-s + (−0.142 + 0.989i)16-s + (0.142 − 0.989i)18-s + 0.563i·19-s + (−1.07 + 0.153i)21-s + (0.544 − 1.19i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
Sign: $-0.654 + 0.755i$
Analytic conductor: \(1.00012\)
Root analytic conductor: \(1.00006\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2004} (2003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2004,\ (\ :0),\ -0.654 + 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4993043540\)
\(L(\frac12)\) \(\approx\) \(0.4993043540\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
3 \( 1 + (-0.142 - 0.989i)T \)
167 \( 1 + T \)
good5 \( 1 + T^{2} \)
7 \( 1 - 1.08iT - T^{2} \)
11 \( 1 + 1.30T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 0.563iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.08iT - T^{2} \)
31 \( 1 - 1.81iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.91T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.284T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.97iT - T^{2} \)
97 \( 1 + 1.30T + T^{2} \)
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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

−9.759174126663420623644662078084, −8.990760292546652808645085232944, −8.231008550869322510560449692963, −7.88125409107015435697786148237, −6.60091962075003501603197464497, −5.66179504088100246065520470711, −5.29706468293422336159705943483, −4.42861349106398789469046077375, −3.25066195071227164316609171247, −2.11363034353980978701349841738, 0.38806655712339262757073986285, 1.71546324256755317324806719742, 2.67145687648998611904777580915, 3.55569975002261590255278865867, 4.61245837670954108584766958980, 5.60918887089368064700192313716, 6.79945178405744898941200331591, 7.61533245718069744622370119429, 7.927032225464557485349158712787, 8.813641439040626230599149037147

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