Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 167 $
Sign $0.841 - 0.540i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.142 + 0.989i)3-s + (−0.654 − 0.755i)4-s + (0.841 + 0.540i)6-s + 1.08i·7-s + (−0.959 + 0.281i)8-s + (−0.959 − 0.281i)9-s + 1.30·11-s + (0.841 − 0.540i)12-s + (0.983 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)18-s + 0.563i·19-s + (−1.07 − 0.153i)21-s + (0.544 − 1.19i)22-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.142 + 0.989i)3-s + (−0.654 − 0.755i)4-s + (0.841 + 0.540i)6-s + 1.08i·7-s + (−0.959 + 0.281i)8-s + (−0.959 − 0.281i)9-s + 1.30·11-s + (0.841 − 0.540i)12-s + (0.983 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)18-s + 0.563i·19-s + (−1.07 − 0.153i)21-s + (0.544 − 1.19i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $0.841 - 0.540i$
motivic weight  =  \(0\)
character  :  $\chi_{2004} (2003, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2004,\ (\ :0),\ 0.841 - 0.540i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.190431511\)
\(L(\frac12)\)  \(\approx\)  \(1.190431511\)
\(L(1)\)   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 \( 1 + (-0.415 + 0.909i)T \)
3 \( 1 + (0.142 - 0.989i)T \)
167 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( 1 - 1.08iT - T^{2} \)
11 \( 1 - 1.30T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 0.563iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.08iT - T^{2} \)
31 \( 1 - 1.81iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.91T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.284T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.97iT - T^{2} \)
97 \( 1 + 1.30T + T^{2} \)
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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

−9.373198838361257665282238493173, −9.013196219139209236541804679348, −8.361252873877966315066126284997, −6.76464643771582641455260546907, −5.82707997005026235943988095459, −5.35484805609351314044853233179, −4.35009334602545203064514680570, −3.64255122552273147148191816058, −2.79893431239952689024205794906, −1.58219327958329133255244849674, 0.821356546807012871468183666687, 2.39591821652881302389623929964, 3.80668576115319999058876370672, 4.29228105616542668826509004898, 5.58508673427531047986895518467, 6.25130226029610824114877687755, 6.92353123945210441463524168047, 7.54108521378009775109095094792, 8.132576618358243388597939307112, 9.099945123526947744198126803519

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