Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 167 $
Sign $-0.142 - 0.989i$
Motivic weight 0
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

\( d \)  =  \(2\)
\( N \)  =  \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-0.142 - 0.989i$
motivic weight  =  \(0\)
character  :  $\chi_{2004} (2003, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2004,\ (\ :0),\ -0.142 - 0.989i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5148868228\)
\(L(\frac12)\)  \(\approx\)  \(0.5148868228\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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