Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 167 $
Sign $-0.654 - 0.755i$
Motivic weight 0
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 + (−0.654 + 0.755i)6-s + 1.51i·7-s + (0.415 − 0.909i)8-s + (0.415 + 0.909i)9-s + 1.91·11-s + (−0.654 − 0.755i)12-s + (−1.49 − 0.215i)14-s + (0.841 + 0.540i)16-s + (−0.959 + 0.281i)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 + (−0.654 + 0.755i)6-s + 1.51i·7-s + (0.415 − 0.909i)8-s + (0.415 + 0.909i)9-s + 1.91·11-s + (−0.654 − 0.755i)12-s + (−1.49 − 0.215i)14-s + (0.841 + 0.540i)16-s + (−0.959 + 0.281i)18-s − 1.81i·19-s + (−0.817 + 1.27i)21-s + (−0.273 + 1.89i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-0.654 - 0.755i$
motivic weight  =  \(0\)
character  :  $\chi_{2004} (2003, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2004,\ (\ :0),\ -0.654 - 0.755i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.430650009\)
\(L(\frac12)\)  \(\approx\)  \(1.430650009\)
\(L(1)\)   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 \( 1 + (0.142 - 0.989i)T \)
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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.195763714498531314714504022990, −9.026060903552162580174836380610, −8.267910459484283156478257617617, −7.31722973824093240994681937910, −6.50732065089162848271645693846, −5.73008358938635899917073877126, −4.80526574760662761237900985092, −4.06231288579216806573854395885, −3.03006468940317916439329839449, −1.77534703422381633789423465072, 1.19323653369796004732161807247, 1.76741432591720677145242629113, 3.33141579554695685553270990557, 3.85049662339810238304695147931, 4.40455664461373128219772250428, 6.05348827826622051144311936367, 6.88411230560012257326069125066, 7.74542586532691781422525631711, 8.303829557899426716957138633203, 9.218210375944351203552894297581

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