Properties

Label 2-2004-2004.2003-c0-0-21
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  + (−0.142 − 0.989i)2-s + (0.841 − 0.540i)3-s + (−0.959 + 0.281i)4-s + (−0.654 − 0.755i)6-s − 1.51i·7-s + (0.415 + 0.909i)8-s + (0.415 − 0.909i)9-s + 1.91·11-s + (−0.654 + 0.755i)12-s + (−1.49 + 0.215i)14-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)18-s + 1.81i·19-s + (−0.817 − 1.27i)21-s + (−0.273 − 1.89i)22-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (0.841 − 0.540i)3-s + (−0.959 + 0.281i)4-s + (−0.654 − 0.755i)6-s − 1.51i·7-s + (0.415 + 0.909i)8-s + (0.415 − 0.909i)9-s + 1.91·11-s + (−0.654 + 0.755i)12-s + (−1.49 + 0.215i)14-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)18-s + 1.81i·19-s + (−0.817 − 1.27i)21-s + (−0.273 − 1.89i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.430650009\)
\(L(\frac12)\) \(\approx\) \(1.430650009\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 + 0.989i)T \)
3 \( 1 + (-0.841 + 0.540i)T \)
167 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( 1 + 1.51iT - T^{2} \)
11 \( 1 - 1.91T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.81iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.51iT - T^{2} \)
31 \( 1 - 1.97iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 0.830T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.68T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.08iT - T^{2} \)
97 \( 1 + 1.91T + T^{2} \)
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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

−9.218210375944351203552894297581, −8.303829557899426716957138633203, −7.74542586532691781422525631711, −6.88411230560012257326069125066, −6.05348827826622051144311936367, −4.40455664461373128219772250428, −3.85049662339810238304695147931, −3.33141579554695685553270990557, −1.76741432591720677145242629113, −1.19323653369796004732161807247, 1.77534703422381633789423465072, 3.03006468940317916439329839449, 4.06231288579216806573854395885, 4.80526574760662761237900985092, 5.73008358938635899917073877126, 6.50732065089162848271645693846, 7.31722973824093240994681937910, 8.267910459484283156478257617617, 9.026060903552162580174836380610, 9.195763714498531314714504022990

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