# Properties

 Degree 1 Conductor $5 \cdot 11 \cdot 19$ Sign $1$ Motivic weight 0 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## 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