Properties

Degree 1
Conductor 1033
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

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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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1033\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{1033} (1032, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 1033,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.963929616$
$L(\frac12,\chi)$  $\approx$  $3.963929616$
$L(\chi,1)$  $\approx$  2.517852251
$L(1,\chi)$  $\approx$  2.517852251

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

−21.45808382779393398213227846294, −20.67457407969200477224594835670, −20.33174312133962866920159905894, −19.43244490890856913209032534802, −18.77154690479651963394004286002, −17.76396403603661915339303261028, −16.40246563763102931932098166676, −15.85755531953847012337881879946, −15.00352710771884096250618920707, −14.54419794513183726709919201147, −13.87428688274220257027142141237, −12.868772019265497311226088206841, −12.167368479151846916339151943681, −11.45407698832281336874744781153, −10.487172839864718332618844564739, −9.61033125669792573566200941329, −8.11895886409747040562712658966, −7.728279079810200086134797658569, −7.28552194238342835712097160231, −5.71768142066804416362431698172, −4.77062758128312056759948333089, −4.19469010108334357398078023047, −3.13297778280283103907163159543, −2.496190219884075234644007637956, −1.32333741042229183244633326411, 1.32333741042229183244633326411, 2.496190219884075234644007637956, 3.13297778280283103907163159543, 4.19469010108334357398078023047, 4.77062758128312056759948333089, 5.71768142066804416362431698172, 7.28552194238342835712097160231, 7.728279079810200086134797658569, 8.11895886409747040562712658966, 9.61033125669792573566200941329, 10.487172839864718332618844564739, 11.45407698832281336874744781153, 12.167368479151846916339151943681, 12.868772019265497311226088206841, 13.87428688274220257027142141237, 14.54419794513183726709919201147, 15.00352710771884096250618920707, 15.85755531953847012337881879946, 16.40246563763102931932098166676, 17.76396403603661915339303261028, 18.77154690479651963394004286002, 19.43244490890856913209032534802, 20.33174312133962866920159905894, 20.67457407969200477224594835670, 21.45808382779393398213227846294

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