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Properties

Label 2-2004-2004.2003-c0-0-13
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  + (0.142 − 0.989i)2-s + (−0.841 + 0.540i)3-s + (−0.959 − 0.281i)4-s + (0.415 + 0.909i)6-s + 1.51i·7-s + (−0.415 + 0.909i)8-s + (0.415 − 0.909i)9-s − 1.91·11-s + (0.959 − 0.281i)12-s + (1.49 + 0.215i)14-s + (0.841 + 0.540i)16-s + (−0.841 − 0.540i)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 + (0.415 + 0.909i)6-s + 1.51i·7-s + (−0.415 + 0.909i)8-s + (0.415 − 0.909i)9-s − 1.91·11-s + (0.959 − 0.281i)12-s + (1.49 + 0.215i)14-s + (0.841 + 0.540i)16-s + (−0.841 − 0.540i)18-s − 1.81i·19-s + (−0.817 − 1.27i)21-s + (−0.273 + 1.89i)22-s + ⋯

Functional equation

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

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

\(L(\frac{1}{2})\) \(\approx\) \(0.2709976748\)
\(L(\frac12)\) \(\approx\) \(0.2709976748\)
\(L(1)\) 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

−9.429498249640891416349232603207, −8.443232273740814134533040307696, −7.62762734096181417469619864582, −6.10199032950348793546427806738, −5.59797369390279907549930120361, −4.95688353425821818503310816831, −4.15926447630209082871961429122, −2.76933209393718318172506862138, −2.31496234839932843104106855748, −0.21067066427896650737313022323, 1.41114859817826356852797458790, 3.25411460284426448719575499226, 4.28388387351064876737646817396, 5.14675861266627290984569925382, 5.70201818103537744369422476957, 6.65292335070755628596519847894, 7.43044524173075439652907654003, 7.73924828491116710202652371896, 8.496527941511592310680743488880, 9.915411180569273505460833811433

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