Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 167 $
Sign $0.415 + 0.909i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.654 + 0.755i)2-s + (−0.959 + 0.281i)3-s + (−0.142 − 0.989i)4-s + (0.415 − 0.909i)6-s − 1.81i·7-s + (0.841 + 0.540i)8-s + (0.841 − 0.540i)9-s + 0.284·11-s + (0.415 + 0.909i)12-s + (1.37 + 1.19i)14-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)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 + (0.415 − 0.909i)6-s − 1.81i·7-s + (0.841 + 0.540i)8-s + (0.841 − 0.540i)9-s + 0.284·11-s + (0.415 + 0.909i)12-s + (1.37 + 1.19i)14-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)18-s + 1.08i·19-s + (0.512 + 1.74i)21-s + (−0.186 + 0.215i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $0.415 + 0.909i$
motivic weight  =  \(0\)
character  :  $\chi_{2004} (2003, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2004,\ (\ :0),\ 0.415 + 0.909i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.4540255234\)
\(L(\frac12)\)  \(\approx\)  \(0.4540255234\)
\(L(1)\)   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 \( 1 + (0.654 - 0.755i)T \)
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 \( 1 + 0.284T + T^{2} \)
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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

−9.566974651866522328012902883976, −8.055842989620565125322608142991, −7.69708109378563967813772693363, −6.78315708564438332244515900693, −6.22248138513001112385307602032, −5.41263286665319115717313654060, −4.26286794105143852689646343218, −3.94336142677950037292621403080, −1.67234156056442930718178238898, −0.48935273346476753254294368942, 1.47650705043448272081151416953, 2.40030282703516699244742858961, 3.43062018570299521503068427378, 4.82603252184221225210623853650, 5.36381234970157321451240062684, 6.46534286212368909642990301565, 7.07702154867784890425600563857, 8.193300695907305300828805979718, 8.822158703955563031513139736901, 9.476222254481463939866979989543

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