Properties

Degree 2
Conductor $ 5 \cdot 17 $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$\zeta_K(s)$  = 1  + 2·3-s + 4-s + 5-s + 2·7-s + 3·9-s + 2·12-s + 2·15-s + 16-s + 17-s + 2·19-s + 20-s + 4·21-s + 2·23-s + 25-s + 4·27-s + 2·28-s + 2·35-s + 3·36-s + 2·37-s + 3·45-s + 2·48-s + 3·49-s + 2·51-s + 4·57-s + 2·59-s + 2·60-s + 6·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  \((2,\ 85,\ (0, 0:\ ),\ 1)\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{85}(84, \cdot))\)

Particular Values

\[\zeta_K(1/2) \approx -1.342635924\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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