Properties

Label 2-2004-2004.2003-c0-0-20
Degree $2$
Conductor $2004$
Sign $-0.142 + 0.989i$
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  + (−0.959 − 0.281i)2-s + (0.415 + 0.909i)3-s + (0.841 + 0.540i)4-s + (−0.142 − 0.989i)6-s − 1.97i·7-s + (−0.654 − 0.755i)8-s + (−0.654 + 0.755i)9-s − 1.68·11-s + (−0.142 + 0.989i)12-s + (−0.557 + 1.89i)14-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)18-s − 1.51i·19-s + (1.80 − 0.822i)21-s + (1.61 + 0.474i)22-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.415 + 0.909i)3-s + (0.841 + 0.540i)4-s + (−0.142 − 0.989i)6-s − 1.97i·7-s + (−0.654 − 0.755i)8-s + (−0.654 + 0.755i)9-s − 1.68·11-s + (−0.142 + 0.989i)12-s + (−0.557 + 1.89i)14-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)18-s − 1.51i·19-s + (1.80 − 0.822i)21-s + (1.61 + 0.474i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
Sign: $-0.142 + 0.989i$
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: \((2,\ 2004,\ (\ :0),\ -0.142 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5148868228\)
\(L(\frac12)\) \(\approx\) \(0.5148868228\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
3 \( 1 + (-0.415 - 0.909i)T \)
167 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( 1 + 1.97iT - T^{2} \)
11 \( 1 + 1.68T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.51iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.97iT - T^{2} \)
31 \( 1 - 0.563iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.30T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 0.830T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.81iT - T^{2} \)
97 \( 1 - 1.68T + T^{2} \)
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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

−9.302611424241345770355970086252, −8.348953769338626287544343296954, −7.62909206074225577765801013553, −7.31936691943083719936813175683, −6.07488912716716976780414689653, −4.81576449070102669769484784272, −4.08439928970688080009990319060, −3.14691847581597375063419529901, −2.25496321622871098715489189879, −0.44030265948757455079775548392, 1.74375930733563195692064760898, 2.42271863402911549332038774359, 3.19567922589099793609408758770, 5.36277631470945834837586314826, 5.67629876629311926280097224196, 6.45834256057662031163942774033, 7.60562711708745556745353334204, 7.989912278624246693271542861769, 8.688740748717579136383507981997, 9.243370031524866655609397340408

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