Properties

Degree 1
Conductor $ 5 \cdot 11 \cdot 19 $
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 − 6-s + 7-s − 8-s + 9-s + 12-s − 13-s − 14-s + 16-s + 17-s − 18-s + 21-s − 23-s − 24-s + 26-s + 27-s + 28-s + 29-s − 31-s − 32-s − 34-s + 36-s + 37-s − 39-s + 41-s + ⋯
L(s,χ)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s − 13-s − 14-s + 16-s + 17-s − 18-s + 21-s − 23-s − 24-s + 26-s + 27-s + 28-s + 29-s − 31-s − 32-s − 34-s + 36-s + 37-s − 39-s + 41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1045\)    =    \(5 \cdot 11 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{1045} (1044, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 1045,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.635030625$
$L(\frac12,\chi)$  $\approx$  $1.635030625$
$L(\chi,1)$  $\approx$  1.132101619
$L(1,\chi)$  $\approx$  1.132101619

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

−21.30723621187722643428886238116, −20.595095077484943238239596582555, −19.82751625309575683545823048511, −19.35816659286193409060258896254, −18.31657383622425928214176155920, −17.93329571795847741735044032882, −16.89300672451798079274714166037, −16.15508739501892846330661532778, −15.19872774539021180981353834881, −14.55610468482603603046112626021, −14.01785228587880129295060213688, −12.60232669165728615618304049872, −12.0036000947278337129896913272, −10.95001475736373849873783436404, −10.098371436516175758172201267262, −9.45391331242561928641935457592, −8.59749343606639663331373171069, −7.66563745787259578805685599829, −7.56929590638183264232915705025, −6.225268278116929425298914412779, −5.03932646373993168302293034692, −3.91991528553599080000997397346, −2.75567243030322196957325020494, −2.04460835684569280896362943688, −1.07136547264467266957227068153, 1.07136547264467266957227068153, 2.04460835684569280896362943688, 2.75567243030322196957325020494, 3.91991528553599080000997397346, 5.03932646373993168302293034692, 6.225268278116929425298914412779, 7.56929590638183264232915705025, 7.66563745787259578805685599829, 8.59749343606639663331373171069, 9.45391331242561928641935457592, 10.098371436516175758172201267262, 10.95001475736373849873783436404, 12.0036000947278337129896913272, 12.60232669165728615618304049872, 14.01785228587880129295060213688, 14.55610468482603603046112626021, 15.19872774539021180981353834881, 16.15508739501892846330661532778, 16.89300672451798079274714166037, 17.93329571795847741735044032882, 18.31657383622425928214176155920, 19.35816659286193409060258896254, 19.82751625309575683545823048511, 20.595095077484943238239596582555, 21.30723621187722643428886238116

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