Properties

Degree $2$
Conductor $191$
Sign $1$
Motivic weight $0$
Arithmetic yes
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  − 1.94·2-s + 1.13·3-s + 2.77·4-s + 0.241·5-s − 2.20·6-s − 3.43·8-s + 0.290·9-s − 0.468·10-s + 3.14·12-s − 1.49·13-s + 0.273·15-s + 3.90·16-s + 1.77·17-s − 0.564·18-s + 0.667·20-s − 0.709·23-s − 3.90·24-s − 0.941·25-s + 2.90·26-s − 0.805·27-s − 0.531·30-s − 4.14·32-s − 3.43·34-s + 0.805·36-s − 1.70·39-s − 0.829·40-s − 0.709·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(191\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 191,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 0.4357716737\] \[L(1,\rho) \approx 0.5984591965\]

Euler product

\(L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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