Properties

Degree 1
Conductor 1021
Sign $-0.999 - 0.0403i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.343 + 0.938i)2-s + (0.932 − 0.361i)3-s + (−0.763 + 0.645i)4-s + (−0.320 + 0.947i)5-s + (0.659 + 0.751i)6-s + (−0.956 − 0.291i)7-s + (−0.869 − 0.494i)8-s + (0.739 − 0.673i)9-s + (−0.999 + 0.0246i)10-s + (0.923 − 0.384i)11-s + (−0.478 + 0.878i)12-s + (0.273 + 0.961i)13-s + (−0.0554 − 0.998i)14-s + (0.0431 + 0.999i)15-s + (0.165 − 0.986i)16-s + (−0.678 + 0.734i)17-s + ⋯
L(s,χ)  = 1  + (0.343 + 0.938i)2-s + (0.932 − 0.361i)3-s + (−0.763 + 0.645i)4-s + (−0.320 + 0.947i)5-s + (0.659 + 0.751i)6-s + (−0.956 − 0.291i)7-s + (−0.869 − 0.494i)8-s + (0.739 − 0.673i)9-s + (−0.999 + 0.0246i)10-s + (0.923 − 0.384i)11-s + (−0.478 + 0.878i)12-s + (0.273 + 0.961i)13-s + (−0.0554 − 0.998i)14-s + (0.0431 + 0.999i)15-s + (0.165 − 0.986i)16-s + (−0.678 + 0.734i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.999 - 0.0403i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.999 - 0.0403i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $-0.999 - 0.0403i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (141, \cdot )$
Sato-Tate  :  $\mu(510)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ -0.999 - 0.0403i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.02647034024 + 1.311106819i$
$L(\frac12,\chi)$  $\approx$  $0.02647034024 + 1.311106819i$
$L(\chi,1)$  $\approx$  0.9125278133 + 0.7801755206i
$L(1,\chi)$  $\approx$  0.9125278133 + 0.7801755206i

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

−21.0254275973477647938184790786, −20.19342566351693146783425881729, −19.9018128922272459871678631930, −19.36004228090852966570730569219, −18.36649388029101301217820456835, −17.40585521558936215566685012570, −16.296175595413375030756835622361, −15.42894553297151599708637144926, −15.034631008713359821903073634158, −13.642971350808042624565709995, −13.37483692473692116034198046416, −12.52361400698107493177567310602, −11.80534637061529684064778787315, −10.77537947098609953192125098095, −9.63029526089360971839659330170, −9.35398225283547336605372192842, −8.62479128356825652116613183129, −7.59926553988988756481154482255, −6.24146716999662844822039067374, −5.17225326025554225653647883570, −4.249051371351435826050686369862, −3.675793991530478228542054261383, −2.6942467916100230831856497064, −1.80258564034104142372000602220, −0.421004004467994221220587594634, 1.67332002473654771614302936625, 3.04887799583674536500541257105, 3.73697169803543088789988950967, 4.21531875556577594582039719851, 6.09876422180649007007738651010, 6.57730441924742205570922847757, 7.115128336166930900918768139631, 8.14395311566649133496065321951, 8.872665931287398332497448209491, 9.6560603996302082408455488803, 10.668400512881423806437747970749, 12.03006028864094690713983936733, 12.628531757299762893625382596833, 13.72866932491041659903787832650, 14.130921469895109484298805895735, 14.747407059196716950491956291321, 15.6477890410474977797321379541, 16.2895291463953239389171012067, 17.17013676195825832988152694743, 18.26955375844024753964897938360, 18.85295003339888201136475688691, 19.46023489548427059162928971296, 20.29987266942114991092216873964, 21.57344199182555563401881080819, 22.0544886801285126339521461183

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