Properties

Degree 2
Conductor $ 5^{2} \cdot 29^{2} $
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  − 4-s − 9-s + 16-s + 36-s − 49-s + 2·59-s − 64-s + 2·71-s + 81-s + 2·109-s − 121-s − 2·139-s − 144-s + 2·149-s + 2·151-s − 169-s + 2·179-s − 2·181-s + 196-s + 2·199-s − 2·236-s + 2·239-s + 2·241-s + 256-s + 2·281-s − 2·284-s − 289-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\R}(s)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21025\)    =    \(5^{2} \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((2,\ 21025,\ (0, 0:\ ),\ 1)\)

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.6927113197\] \[L(1,\rho) \approx 0.7675955600\]

Imaginary part of the first few zeros on the critical line

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