Properties

Degree 2
Conductor $ 2^{2} \cdot 5^{2} $
Sign $unknown$
Motivic weight 0
Primitive yes
Self-dual no

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)10-s + (−0.5 − 0.363i)13-s + (−0.809 − 0.587i)16-s + (−0.500 − 1.53i)17-s + 0.999·18-s + 0.999·20-s + (−0.809 + 0.587i)25-s + 0.618·26-s + (−0.499 + 1.53i)29-s + 0.999·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ unknown}) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100\)    =    \(2^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $unknown$
primitive  :  yes
self-dual  :  no
Selberg data  =  \((2,\ 100,\ (0, 1:\ ),\ 0)\)

Euler product

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

Particular Values

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

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.