Properties

Degree 2
Conductor 2011
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 0.445·5-s + 9-s + 1.24·13-s + 16-s − 0.445·20-s − 1.80·23-s − 0.801·25-s − 1.80·31-s + 36-s + 1.24·41-s + 1.24·43-s − 0.445·45-s + 49-s + 1.24·52-s + 64-s − 0.554·65-s + 1.24·71-s − 0.445·80-s + 81-s − 1.80·83-s − 1.80·89-s − 1.80·92-s − 0.801·100-s − 0.445·101-s − 0.445·103-s − 0.445·109-s + ⋯
L(s)  = 1  + 4-s − 0.445·5-s + 9-s + 1.24·13-s + 16-s − 0.445·20-s − 1.80·23-s − 0.801·25-s − 1.80·31-s + 36-s + 1.24·41-s + 1.24·43-s − 0.445·45-s + 49-s + 1.24·52-s + 64-s − 0.554·65-s + 1.24·71-s − 0.445·80-s + 81-s − 1.80·83-s − 1.80·89-s − 1.80·92-s − 0.801·100-s − 0.445·101-s − 0.445·103-s − 0.445·109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2011\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2011} (2010, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2011,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.504006919$
$L(\frac12)$  $\approx$  $1.504006919$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2011$, \(F_p\) is a polynomial of degree 2. If $p = 2011$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2011 \( 1+O(T) \)
good2 \( 1 - T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + 0.445T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.24T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.80T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.80T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.24T + T^{2} \)
43 \( 1 - 1.24T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.24T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.80T + T^{2} \)
89 \( 1 + 1.80T + T^{2} \)
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

−9.433165867166389605100889166681, −8.383017311307391300186212949089, −7.64499294080361593415047298585, −7.13914371461768728472912077340, −6.14374611519011938166547855601, −5.63800325129470412524547640051, −4.07556281147085956434886199465, −3.75370341881959017979953493370, −2.34543493833801924716716633503, −1.40324453744509293915956313039, 1.40324453744509293915956313039, 2.34543493833801924716716633503, 3.75370341881959017979953493370, 4.07556281147085956434886199465, 5.63800325129470412524547640051, 6.14374611519011938166547855601, 7.13914371461768728472912077340, 7.64499294080361593415047298585, 8.383017311307391300186212949089, 9.433165867166389605100889166681

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