Properties

Degree 1
Conductor 1021
Sign $-0.517 + 0.855i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.510 − 0.859i)2-s + (−0.602 + 0.798i)3-s + (−0.478 − 0.878i)4-s + (0.378 + 0.925i)5-s + (0.378 + 0.925i)6-s + (−0.201 − 0.979i)7-s + (−0.999 − 0.0369i)8-s + (−0.273 − 0.961i)9-s + (0.989 + 0.147i)10-s + (−0.713 + 0.700i)11-s + (0.989 + 0.147i)12-s + (0.0922 − 0.995i)13-s + (−0.945 − 0.326i)14-s + (−0.966 − 0.255i)15-s + (−0.542 + 0.840i)16-s + (0.237 − 0.971i)17-s + ⋯
L(s,χ)  = 1  + (0.510 − 0.859i)2-s + (−0.602 + 0.798i)3-s + (−0.478 − 0.878i)4-s + (0.378 + 0.925i)5-s + (0.378 + 0.925i)6-s + (−0.201 − 0.979i)7-s + (−0.999 − 0.0369i)8-s + (−0.273 − 0.961i)9-s + (0.989 + 0.147i)10-s + (−0.713 + 0.700i)11-s + (0.989 + 0.147i)12-s + (0.0922 − 0.995i)13-s + (−0.945 − 0.326i)14-s + (−0.966 − 0.255i)15-s + (−0.542 + 0.840i)16-s + (0.237 − 0.971i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.517 + 0.855i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.517 + 0.855i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $-0.517 + 0.855i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (125, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ -0.517 + 0.855i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1192795766 + 0.2115600828i$
$L(\frac12,\chi)$  $\approx$  $0.1192795766 + 0.2115600828i$
$L(\chi,1)$  $\approx$  0.7956685758 - 0.1568609050i
$L(1,\chi)$  $\approx$  0.7956685758 - 0.1568609050i

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

−21.54689663728212701621748439887, −21.02066655142834291427893133350, −19.49204933933706208537928469674, −18.8903596988694749929652404240, −17.94700623646365255040456523503, −17.37024482207516755420608476990, −16.55956379569506740911089110679, −16.00751183009565573676706572436, −15.18627801195923225903048570594, −14.00688250810800830104375309382, −13.226172853854332052722231824548, −12.94326580891777663769300127291, −11.894350735437243937422536905, −11.47273806615539242073771937182, −9.89267564892300096352032220918, −8.80290733870139002906457184776, −8.33485310244108012907786033638, −7.408725270510753160241279798858, −6.17224891794727960572024111099, −5.921295185044308111562404369462, −5.09887250727321326682036537756, −4.18808415124643482762908937801, −2.7135793636719694711552868595, −1.75178029355818873527216775439, −0.09162295587398586398210970059, 1.38405592510436906507973304033, 2.86605065325229341831144212627, 3.3369491478535432855821531797, 4.44925208807366824155435966009, 5.13903686170255296219100341385, 6.110505224714665526549976711348, 6.875198685340899776535602068323, 8.13467597450958235395991539126, 9.66454899550360703328405408048, 10.14593352395354381424953804581, 10.510459005562752751488849431167, 11.30363993760783082390029981517, 12.26421989019242042502507604678, 13.06551695264263180246550305017, 14.00026084543860993766247372709, 14.68097028945002760130720372246, 15.38055458941533931421973363760, 16.318571965953553175493572826182, 17.31782269113509498699155919246, 18.211091373526829277974567403204, 18.48978891143399285692867273229, 19.981407879087002854920404537443, 20.389759814789903260349558282365, 21.09622621488669403625038695686, 21.92426457627678040402612770882

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