Properties

Degree 2
Conductor 148
Sign $1$
Self-dual yes
Motivic weight 0

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  − 3-s − 7-s − 11-s + 21-s + 25-s + 27-s + 33-s + 37-s − 41-s − 47-s − 53-s + 2·67-s − 71-s − 73-s − 75-s + 77-s − 81-s − 83-s − 101-s + 2·107-s − 111-s + 123-s − 127-s + 2·137-s + 2·139-s + 141-s − 149-s + ⋯

Functional equation

\[\begin{align} \Lambda(s)=\mathstrut & 148 ^{s/2} \Gamma_{\R}(s) ^{2} \cdot L(s,\rho)\cr =\mathstrut & \Lambda(1-s) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(148\)    =    \(2^{2} \cdot 37\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 148,\ (0, 0:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.2355320219\] \[L(1,\rho) \approx 0.5465728811\]

Imaginary part of the first few zeros on the critical line

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