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 + 1.24·5-s + 9-s − 1.80·13-s + 16-s + 1.24·20-s − 0.445·23-s + 0.554·25-s − 0.445·31-s + 36-s − 1.80·41-s − 1.80·43-s + 1.24·45-s + 49-s − 1.80·52-s + 64-s − 2.24·65-s − 1.80·71-s + 1.24·80-s + 81-s − 0.445·83-s − 0.445·89-s − 0.445·92-s + 0.554·100-s + 1.24·101-s + 1.24·103-s + 1.24·109-s + ⋯
L(s)  = 1  + 4-s + 1.24·5-s + 9-s − 1.80·13-s + 16-s + 1.24·20-s − 0.445·23-s + 0.554·25-s − 0.445·31-s + 36-s − 1.80·41-s − 1.80·43-s + 1.24·45-s + 49-s − 1.80·52-s + 64-s − 2.24·65-s − 1.80·71-s + 1.24·80-s + 81-s − 0.445·83-s − 0.445·89-s − 0.445·92-s + 0.554·100-s + 1.24·101-s + 1.24·103-s + 1.24·109-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  =  $(2,\ 2011,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.791490749$
$L(\frac12)$  $\approx$  $1.791490749$
$L(1)$   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 \( 1 - 1.24T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.80T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 0.445T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.80T + T^{2} \)
43 \( 1 + 1.80T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.80T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.445T + T^{2} \)
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

−9.745431022933099867967448078128, −8.586881907403457608406430015799, −7.45881190023126099539701954613, −7.05296604711855684608828411781, −6.25947800722014296123538162185, −5.41764417430903220491527889275, −4.65219973318233909750970594027, −3.28691773354206112475033510574, −2.21394021840978122774154295075, −1.67988930515697396102481732042, 1.67988930515697396102481732042, 2.21394021840978122774154295075, 3.28691773354206112475033510574, 4.65219973318233909750970594027, 5.41764417430903220491527889275, 6.25947800722014296123538162185, 7.05296604711855684608828411781, 7.45881190023126099539701954613, 8.586881907403457608406430015799, 9.745431022933099867967448078128

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