Properties

Degree 1
Conductor 1069
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s + 22-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + ⋯
L(s,χ)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s + 22-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1069 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1069 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1069\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{1069} (1068, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 1069,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.631358902$
$L(\frac12,\chi)$  $\approx$  $1.631358902$
$L(\chi,1)$  $\approx$  1.130837073
$L(1,\chi)$  $\approx$  1.130837073

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.335460833047656495140365130206, −20.409322519220527118964950520557, −20.06378608959351325520470004059, −19.008272429606601034939503437, −18.24394416848617688462810098572, −18.11593292037744131507207605725, −16.59240436530444804047272979750, −16.17980026536149033886025146918, −15.4407942548402582478739321225, −14.43328298143321991171241748074, −13.55548301000671350748392684138, −12.972117579010113188385790385892, −12.00982186723874336149593319022, −10.65280932372088875399381707475, −9.932688443406725796468351208004, −9.65339308149452060061459471800, −8.64211878771895071505699268366, −7.97711934290423326724046601309, −7.030901446065347669681940913361, −6.19405389479680696127967076793, −5.33758266153677494701938146712, −3.53854593842573313294799984254, −2.949263370571121883871299420286, −2.0421275606331166992099861817, −1.049324784840336105248480366706, 1.049324784840336105248480366706, 2.0421275606331166992099861817, 2.949263370571121883871299420286, 3.53854593842573313294799984254, 5.33758266153677494701938146712, 6.19405389479680696127967076793, 7.030901446065347669681940913361, 7.97711934290423326724046601309, 8.64211878771895071505699268366, 9.65339308149452060061459471800, 9.932688443406725796468351208004, 10.65280932372088875399381707475, 12.00982186723874336149593319022, 12.972117579010113188385790385892, 13.55548301000671350748392684138, 14.43328298143321991171241748074, 15.4407942548402582478739321225, 16.17980026536149033886025146918, 16.59240436530444804047272979750, 18.11593292037744131507207605725, 18.24394416848617688462810098572, 19.008272429606601034939503437, 20.06378608959351325520470004059, 20.409322519220527118964950520557, 21.335460833047656495140365130206

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