↑

Properties

Label 2-2004-2004.2003-c0-0-25
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.654 − 0.755i)2-s + (0.959 + 0.281i)3-s + (−0.142 − 0.989i)4-s + (0.841 − 0.540i)6-s − 1.81i·7-s + (−0.841 − 0.540i)8-s + (0.841 + 0.540i)9-s − 0.284·11-s + (0.142 − 0.989i)12-s + (−1.37 − 1.19i)14-s + (−0.959 + 0.281i)16-s + (0.959 − 0.281i)18-s + 1.08i·19-s + (0.512 − 1.74i)21-s + (−0.186 + 0.215i)22-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)2-s + (0.959 + 0.281i)3-s + (−0.142 − 0.989i)4-s + (0.841 − 0.540i)6-s − 1.81i·7-s + (−0.841 − 0.540i)8-s + (0.841 + 0.540i)9-s − 0.284·11-s + (0.142 − 0.989i)12-s + (−1.37 − 1.19i)14-s + (−0.959 + 0.281i)16-s + (0.959 − 0.281i)18-s + 1.08i·19-s + (0.512 − 1.74i)21-s + (−0.186 + 0.215i)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

\(L(\frac{1}{2})\) \(\approx\) \(2.115157431\)
\(L(\frac12)\) \(\approx\) \(2.115157431\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 + 0.755i)T \)
3 \( 1 + (-0.959 - 0.281i)T \)
167 \( 1 + T \)
good5 \( 1 + T^{2} \)
7 \( 1 + 1.81iT - T^{2} \)
11 \( 1 + 0.284T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.08iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.81iT - T^{2} \)
31 \( 1 + 1.51iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.68T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.91T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.563iT - T^{2} \)
97 \( 1 + 0.284T + T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.456077795634699530101641341005, −8.382994765529261989796640573018, −7.55120582527248838673955873961, −6.92350573513592376744364707005, −5.74314992456526547165499965682, −4.70306378913871088043956747517, −3.88235652609550410901517400954, −3.57128759328048255234779438564, −2.31161387586290419410656768922, −1.22245328775311170575418516194, 2.25244002593004489515115139250, 2.70863447766159891334552272495, 3.77265474540042475412083956624, 4.80961839359736225721455948563, 5.64625017279327086286159003177, 6.36217766714101534995888933778, 7.23285925251120084198840539838, 8.044079583740304294688288077683, 8.664992887661533317813245597190, 9.143377215444846548352808850398

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