Properties

Degree 1
Conductor 1021
Sign $0.832 + 0.554i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.412 + 0.911i)2-s + (0.445 + 0.895i)3-s + (−0.659 − 0.751i)4-s + (−0.999 + 0.0369i)5-s + (−0.999 + 0.0369i)6-s + (−0.0554 + 0.998i)7-s + (0.956 − 0.291i)8-s + (−0.602 + 0.798i)9-s + (0.378 − 0.925i)10-s + (0.997 − 0.0738i)11-s + (0.378 − 0.925i)12-s + (0.739 − 0.673i)13-s + (−0.886 − 0.462i)14-s + (−0.478 − 0.878i)15-s + (−0.128 + 0.991i)16-s + (−0.343 − 0.938i)17-s + ⋯
L(s,χ)  = 1  + (−0.412 + 0.911i)2-s + (0.445 + 0.895i)3-s + (−0.659 − 0.751i)4-s + (−0.999 + 0.0369i)5-s + (−0.999 + 0.0369i)6-s + (−0.0554 + 0.998i)7-s + (0.956 − 0.291i)8-s + (−0.602 + 0.798i)9-s + (0.378 − 0.925i)10-s + (0.997 − 0.0738i)11-s + (0.378 − 0.925i)12-s + (0.739 − 0.673i)13-s + (−0.886 − 0.462i)14-s + (−0.478 − 0.878i)15-s + (−0.128 + 0.991i)16-s + (−0.343 − 0.938i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.832 + 0.554i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.832 + 0.554i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $0.832 + 0.554i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (91, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ 0.832 + 0.554i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8338129818 + 0.2523838531i$
$L(\frac12,\chi)$  $\approx$  $0.8338129818 + 0.2523838531i$
$L(\chi,1)$  $\approx$  0.6716775749 + 0.4164441638i
$L(1,\chi)$  $\approx$  0.6716775749 + 0.4164441638i

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

−21.22560934327880135514586579358, −20.440569112549225700918329030699, −19.76882443569245220236388503465, −19.37144343667721138150030296413, −18.73019239276765538426464218377, −17.80048848451587785313787822576, −16.97339928954737887039583214284, −16.40869308680128061023772281936, −14.97878387487787558565575048915, −14.23067832340099908129925203575, −13.36264325874465176212138615219, −12.78268226679564800566362675732, −11.77161676232505625542144794118, −11.40936371406869015002563151189, −10.42220263130242901633709091055, −9.33817992656486172594723843391, −8.49579011289939700248181839384, −7.93508089704298516552246377067, −7.06706917012410267735790616307, −6.26975313165334298051305818140, −4.33478666257074703901035242778, −3.87111694634618200280829610018, −3.11351683762612511955977337661, −1.60168171729647672517260227015, −1.16823954540136066310218921868, 0.47285857224140128255418628572, 2.31296638028646150528187231456, 3.48819171689372816922721568057, 4.36221368912139204277616526292, 5.11430626170110149445899742379, 6.14339355808368681942305128734, 7.04338179738133832662327453860, 8.17390156477124880562886005578, 8.73332211191318421564174163570, 9.17864501934940368916449575001, 10.31208241327160957011713904866, 11.12479684576213413093278478354, 11.97087333139227586386041586870, 13.18767198978686573675918513788, 14.18670213696434694777572749812, 14.882507708761512889167472947783, 15.563196542695770417651630314137, 15.89657157111214563790969227247, 16.7210972094733754209360791131, 17.68992834794238016913785550662, 18.58373446080008660729892617116, 19.36292224666707548931119130856, 19.87873783147964941741887112573, 20.81649814238577846117053301187, 21.88645398109524887294416013097

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