Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 167 $
Sign $-0.959 + 0.281i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.654 − 0.755i)3-s + (0.415 − 0.909i)4-s + (−0.959 − 0.281i)6-s − 0.563i·7-s + (−0.142 − 0.989i)8-s + (−0.142 + 0.989i)9-s − 0.830·11-s + (−0.959 + 0.281i)12-s + (−0.304 − 0.474i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)18-s − 1.97i·19-s + (−0.425 + 0.368i)21-s + (−0.698 + 0.449i)22-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.654 − 0.755i)3-s + (0.415 − 0.909i)4-s + (−0.959 − 0.281i)6-s − 0.563i·7-s + (−0.142 − 0.989i)8-s + (−0.142 + 0.989i)9-s − 0.830·11-s + (−0.959 + 0.281i)12-s + (−0.304 − 0.474i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)18-s − 1.97i·19-s + (−0.425 + 0.368i)21-s + (−0.698 + 0.449i)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

\( d \)  =  \(2\)
\( N \)  =  \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-0.959 + 0.281i$
motivic weight  =  \(0\)
character  :  $\chi_{2004} (2003, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2004,\ (\ :0),\ -0.959 + 0.281i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.246580327\)
\(L(\frac12)\)  \(\approx\)  \(1.246580327\)
\(L(1)\)   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 \( 1 + (-0.841 + 0.540i)T \)
3 \( 1 + (0.654 + 0.755i)T \)
167 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( 1 + 0.563iT - T^{2} \)
11 \( 1 + 0.830T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.97iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 0.563iT - T^{2} \)
31 \( 1 - 1.08iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 0.284T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.30T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.51iT - T^{2} \)
97 \( 1 - 0.830T + T^{2} \)
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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

−9.111574395020783185049358657916, −7.965273954320971316860709705919, −7.13211985254516484418847843169, −6.62647133182427370300378294441, −5.62377454060395846042048766948, −5.01185791478808765445415106960, −4.18901539981076442664756016075, −2.91223632622808443979817457861, −2.07722669255879354796332111870, −0.71142446747013975391390430563, 2.15659483998477071812008552309, 3.37295572007839185449154365206, 4.07211317649829003086666662675, 5.01124189604859047739106667333, 5.80365273311669929055291981360, 6.04219994701079969276257205102, 7.26822386975117055471082751863, 8.038819135943094733431140174035, 8.795977302632933997219909822602, 9.827644319519802074081448802060

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