Properties

Label 2-2011-2011.2010-c0-0-0
Degree $2$
Conductor $2011$
Sign $1$
Analytic cond. $1.00361$
Root an. cond. $1.00180$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 1.80·5-s + 9-s − 0.445·13-s + 16-s − 1.80·20-s + 1.24·23-s + 2.24·25-s + 1.24·31-s + 36-s − 0.445·41-s − 0.445·43-s − 1.80·45-s + 49-s − 0.445·52-s + 64-s + 0.801·65-s − 0.445·71-s − 1.80·80-s + 81-s + 1.24·83-s + 1.24·89-s + 1.24·92-s + 2.24·100-s − 1.80·101-s − 1.80·103-s − 1.80·109-s + ⋯
L(s)  = 1  + 4-s − 1.80·5-s + 9-s − 0.445·13-s + 16-s − 1.80·20-s + 1.24·23-s + 2.24·25-s + 1.24·31-s + 36-s − 0.445·41-s − 0.445·43-s − 1.80·45-s + 49-s − 0.445·52-s + 64-s + 0.801·65-s − 0.445·71-s − 1.80·80-s + 81-s + 1.24·83-s + 1.24·89-s + 1.24·92-s + 2.24·100-s − 1.80·101-s − 1.80·103-s − 1.80·109-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

Degree: \(2\)
Conductor: \(2011\)
Sign: $1$
Analytic conductor: \(1.00361\)
Root analytic conductor: \(1.00180\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (2010, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2011,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.179929084\)
\(L(\frac12)\) \(\approx\) \(1.179929084\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1+O(T) \)
good2 \( 1 - T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + 1.80T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 0.445T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.24T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.24T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 0.445T + T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 0.445T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.24T + T^{2} \)
89 \( 1 - 1.24T + T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.307637544819940100064989679548, −8.260296202951996999396743795541, −7.71034924230206571734003232675, −7.03384206160308353072004579797, −6.60257240899894917415705156396, −5.17284836000195726489118549613, −4.34831431540597383193760842166, −3.52165873841477642791745100310, −2.64131041767919126497260176640, −1.14161997012900792784024894394, 1.14161997012900792784024894394, 2.64131041767919126497260176640, 3.52165873841477642791745100310, 4.34831431540597383193760842166, 5.17284836000195726489118549613, 6.60257240899894917415705156396, 7.03384206160308353072004579797, 7.71034924230206571734003232675, 8.260296202951996999396743795541, 9.307637544819940100064989679548

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