Properties

Degree 2
Conductor 37
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 + 2·7-s + 3·9-s + 2·11-s + 2·12-s + 16-s + 4·21-s + 25-s + 4·27-s + 2·28-s + 4·33-s + 3·36-s + 37-s + 2·41-s + 2·44-s + 2·47-s + 2·48-s + 3·49-s + 2·53-s + 6·63-s + 64-s + 2·67-s + 2·71-s + 2·73-s + 2·75-s + 4·77-s + ⋯

Functional equation

\[\begin{aligned} \Lambda_K(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(37\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  $(2,\ 37,\ (0, 0:\ ),\ 1)$

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{37}(36, \cdot))\)

Particular Values

\[\zeta_K(1/2) \approx -0.9769616189\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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