Properties

Degree $2$
Conductor $8281$
Sign $unknown$
Motivic weight $0$
Arithmetic yes
Primitive yes
Self-dual no

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + (−0.707 + 0.707i)2-s + 3-s + (0.707 − 0.707i)5-s + (−0.707 + 0.707i)6-s + (−0.707 − 0.707i)8-s + i·10-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)15-s + 16-s + i·17-s + (0.707 − 0.707i)19-s − 22-s + i·23-s + (−0.707 − 0.707i)24-s − 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ unknown}) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8281\)    =    \(7^{2} \cdot 13^{2}\)
Sign: $unknown$
Arithmetic: yes
Primitive: yes
Self-dual: no
Selberg data: \((2,\ 8281,\ (0, 1:\ ),\ 0)\)

Particular Values

Not enough information (Dirichlet series coefficients/sign of the functional equation) to compute special values.

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

Zeros not available.

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

Plot not available.