Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 167 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 1.41·5-s − 6-s − 8-s + 9-s − 1.41·10-s + 12-s + 1.41·15-s + 16-s − 1.41·17-s − 18-s + 1.41·20-s − 24-s + 1.00·25-s + 27-s − 1.41·30-s − 32-s + 1.41·34-s + 36-s − 1.41·40-s − 1.41·41-s + 1.41·43-s + 1.41·45-s + 48-s + 49-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s + 1.41·5-s − 6-s − 8-s + 9-s − 1.41·10-s + 12-s + 1.41·15-s + 16-s − 1.41·17-s − 18-s + 1.41·20-s − 24-s + 1.00·25-s + 27-s − 1.41·30-s − 32-s + 1.41·34-s + 36-s − 1.41·40-s − 1.41·41-s + 1.41·43-s + 1.41·45-s + 48-s + 49-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2004} (2003, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2004,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.354874869$
$L(\frac12)$  $\approx$  $1.354874869$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 - T \)
167 \( 1 - T \)
good5 \( 1 - 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.153185246128506526850677113033, −8.879276186849607427647040622757, −7.987965655892556815219143801371, −7.07082369257477852149992716431, −6.46815055191435980337744499833, −5.59971734959201756791184949995, −4.36522854736731233247351960732, −3.02775246984645232156406584006, −2.25298910163709615521970108812, −1.52787103337297115316766718154, 1.52787103337297115316766718154, 2.25298910163709615521970108812, 3.02775246984645232156406584006, 4.36522854736731233247351960732, 5.59971734959201756791184949995, 6.46815055191435980337744499833, 7.07082369257477852149992716431, 7.987965655892556815219143801371, 8.879276186849607427647040622757, 9.153185246128506526850677113033

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