Properties

Degree 1
Conductor 3089
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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 3089 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 3089 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(3089\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{3089} (3088, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 3089,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.795080609$
$L(\frac12,\chi)$  $\approx$  $3.795080609$
$L(\chi,1)$  $\approx$  2.081935563
$L(1,\chi)$  $\approx$  2.081935563

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

−18.98649425919922027383647560632, −18.004199620609539367537306367470, −17.44351822324532954567558213817, −17.02493818469749755894759255905, −16.27950932735102271842304995026, −15.37364308290442122436725667941, −14.764347829619427936122611578206, −13.88570998871122028088828843038, −13.63144062534870326553588677240, −12.51959333312883130644871120369, −11.98085165623017008934874224365, −11.45913236734416915541628090527, −10.69307990958868372900884136291, −10.01160083113616235693446871198, −9.24187974306513054710002180997, −8.02767727997180390547686379038, −6.97356602603096927401170078286, −6.72920960020158238382711565522, −5.52306668304908549455122699397, −5.438708976340597120582302272595, −4.47067187451388179389619871835, −3.90957553692704579906895026368, −2.47236215421442630510624015700, −1.8512691452178303295978326314, −1.08123634509602035847304545739, 1.08123634509602035847304545739, 1.8512691452178303295978326314, 2.47236215421442630510624015700, 3.90957553692704579906895026368, 4.47067187451388179389619871835, 5.438708976340597120582302272595, 5.52306668304908549455122699397, 6.72920960020158238382711565522, 6.97356602603096927401170078286, 8.02767727997180390547686379038, 9.24187974306513054710002180997, 10.01160083113616235693446871198, 10.69307990958868372900884136291, 11.45913236734416915541628090527, 11.98085165623017008934874224365, 12.51959333312883130644871120369, 13.63144062534870326553588677240, 13.88570998871122028088828843038, 14.764347829619427936122611578206, 15.37364308290442122436725667941, 16.27950932735102271842304995026, 17.02493818469749755894759255905, 17.44351822324532954567558213817, 18.004199620609539367537306367470, 18.98649425919922027383647560632

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