Properties

Degree 2
Conductor $ 3 \cdot 37 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + 3-s − 4-s − 2·7-s + 9-s − 12-s + 16-s − 2·21-s − 25-s + 27-s + 2·28-s − 36-s + 37-s + 48-s + 3·49-s − 2·63-s − 64-s − 2·67-s − 2·73-s − 75-s + 81-s + 2·84-s + 100-s − 108-s + 111-s − 2·112-s + 121-s + 2·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(111\)    =    \(3 \cdot 37\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 111,\ (0, 1:\ ),\ 1)$

Euler product

\[\begin{aligned}L(s,\rho) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 0.5829903212\] \[L(1,\rho) \approx 0.8711715270\]

Imaginary part of the first few zeros on the critical line

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