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Properties

Label 2-2004-2004.2003-c0-0-4
Degree $2$
Conductor $2004$
Sign $-0.959 - 0.281i$
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  + (0.841 + 0.540i)2-s + (−0.654 + 0.755i)3-s + (0.415 + 0.909i)4-s + (−0.959 + 0.281i)6-s + 0.563i·7-s + (−0.142 + 0.989i)8-s + (−0.142 − 0.989i)9-s − 0.830·11-s + (−0.959 − 0.281i)12-s + (−0.304 + 0.474i)14-s + (−0.654 + 0.755i)16-s + (0.415 − 0.909i)18-s + 1.97i·19-s + (−0.425 − 0.368i)21-s + (−0.698 − 0.449i)22-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (−0.654 + 0.755i)3-s + (0.415 + 0.909i)4-s + (−0.959 + 0.281i)6-s + 0.563i·7-s + (−0.142 + 0.989i)8-s + (−0.142 − 0.989i)9-s − 0.830·11-s + (−0.959 − 0.281i)12-s + (−0.304 + 0.474i)14-s + (−0.654 + 0.755i)16-s + (0.415 − 0.909i)18-s + 1.97i·19-s + (−0.425 − 0.368i)21-s + (−0.698 − 0.449i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
Sign: $-0.959 - 0.281i$
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: \((2,\ 2004,\ (\ :0),\ -0.959 - 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246580327\)
\(L(\frac12)\) \(\approx\) \(1.246580327\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 - 0.540i)T \)
3 \( 1 + (0.654 - 0.755i)T \)
167 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( 1 - 0.563iT - T^{2} \)
11 \( 1 + 0.830T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.97iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 0.563iT - T^{2} \)
31 \( 1 + 1.08iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 0.284T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.30T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.51iT - T^{2} \)
97 \( 1 - 0.830T + T^{2} \)
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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

−9.827644319519802074081448802060, −8.795977302632933997219909822602, −8.038819135943094733431140174035, −7.26822386975117055471082751863, −6.04219994701079969276257205102, −5.80365273311669929055291981360, −5.01124189604859047739106667333, −4.07211317649829003086666662675, −3.37295572007839185449154365206, −2.15659483998477071812008552309, 0.71142446747013975391390430563, 2.07722669255879354796332111870, 2.91223632622808443979817457861, 4.18901539981076442664756016075, 5.01185791478808765445415106960, 5.62377454060395846042048766948, 6.62647133182427370300378294441, 7.13211985254516484418847843169, 7.965273954320971316860709705919, 9.111574395020783185049358657916

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