Properties

Degree 2
Conductor $ 2^{3} \cdot 251 $
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 + 4-s − 0.445·7-s + 8-s + 9-s − 1.80·11-s − 0.445·14-s + 16-s + 1.24·17-s + 18-s + 1.24·19-s − 1.80·22-s − 1.80·23-s + 25-s − 0.445·28-s + 1.24·29-s + 1.24·31-s + 32-s + 1.24·34-s + 36-s − 0.445·37-s + 1.24·38-s − 1.80·41-s − 0.445·43-s − 1.80·44-s − 1.80·46-s − 0.801·49-s + ⋯
L(s)  = 1  + 2-s + 4-s − 0.445·7-s + 8-s + 9-s − 1.80·11-s − 0.445·14-s + 16-s + 1.24·17-s + 18-s + 1.24·19-s − 1.80·22-s − 1.80·23-s + 25-s − 0.445·28-s + 1.24·29-s + 1.24·31-s + 32-s + 1.24·34-s + 36-s − 0.445·37-s + 1.24·38-s − 1.80·41-s − 0.445·43-s − 1.80·44-s − 1.80·46-s − 0.801·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2008\)    =    \(2^{3} \cdot 251\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2008} (501, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2008,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(2.189949895\)
\(L(\frac12)\)  \(\approx\)  \(2.189949895\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;251\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
251 \( 1 - T \)
good3 \( 1 - T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + 0.445T + T^{2} \)
11 \( 1 + 1.80T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.24T + T^{2} \)
19 \( 1 - 1.24T + T^{2} \)
23 \( 1 + 1.80T + T^{2} \)
29 \( 1 - 1.24T + T^{2} \)
31 \( 1 - 1.24T + T^{2} \)
37 \( 1 + 0.445T + T^{2} \)
41 \( 1 + 1.80T + T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.80T + T^{2} \)
59 \( 1 + 0.445T + T^{2} \)
61 \( 1 + 1.80T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.445T + T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 0.445T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.787816506003684698192300407786, −8.072614514188752656720750219597, −7.82135070647735025724817902482, −6.85446611413567599809651854315, −6.10152828771663925452510106786, −5.13453697736513187885014052533, −4.68071065382005982257663483019, −3.38788895230886524496922558945, −2.85265783201785653389638142199, −1.51044338409629372437500213854, 1.51044338409629372437500213854, 2.85265783201785653389638142199, 3.38788895230886524496922558945, 4.68071065382005982257663483019, 5.13453697736513187885014052533, 6.10152828771663925452510106786, 6.85446611413567599809651854315, 7.82135070647735025724817902482, 8.072614514188752656720750219597, 9.787816506003684698192300407786

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