Properties

Degree 1
Conductor 311
Sign $0.371 + 0.928i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.994 + 0.101i)2-s + (−0.994 − 0.101i)3-s + (0.979 − 0.201i)4-s + (0.688 + 0.724i)5-s + 6-s + (0.347 − 0.937i)7-s + (−0.954 + 0.299i)8-s + (0.979 + 0.201i)9-s + (−0.758 − 0.651i)10-s + (0.0506 + 0.998i)11-s + (−0.994 + 0.101i)12-s + (0.979 + 0.201i)13-s + (−0.250 + 0.968i)14-s + (−0.612 − 0.790i)15-s + (0.918 − 0.394i)16-s + (0.758 + 0.651i)17-s + ⋯
L(s,χ)  = 1  + (−0.994 + 0.101i)2-s + (−0.994 − 0.101i)3-s + (0.979 − 0.201i)4-s + (0.688 + 0.724i)5-s + 6-s + (0.347 − 0.937i)7-s + (−0.954 + 0.299i)8-s + (0.979 + 0.201i)9-s + (−0.758 − 0.651i)10-s + (0.0506 + 0.998i)11-s + (−0.994 + 0.101i)12-s + (0.979 + 0.201i)13-s + (−0.250 + 0.968i)14-s + (−0.612 − 0.790i)15-s + (0.918 − 0.394i)16-s + (0.758 + 0.651i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.371 + 0.928i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.371 + 0.928i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(311\)
\( \varepsilon \)  =  $0.371 + 0.928i$
motivic weight  =  \(0\)
character  :  $\chi_{311} (206, \cdot )$
Sato-Tate  :  $\mu(62)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 311,\ (1:\ ),\ 0.371 + 0.928i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9048540990 + 0.6122517454i$
$L(\frac12,\chi)$  $\approx$  $0.9048540990 + 0.6122517454i$
$L(\chi,1)$  $\approx$  0.6856489059 + 0.1475121719i
$L(1,\chi)$  $\approx$  0.6856489059 + 0.1475121719i

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

−24.91106051009643791919274780901, −24.21566226567802179651194333637, −23.21634465242036312765870901892, −21.69843909814459726265376864079, −21.346930352542665044630803294472, −20.559036090546870330825827817451, −19.01567526500630559624050198607, −18.49155832269946370191447901227, −17.507773576980350426264489002317, −16.93495765782660380209450065431, −15.991816030368337128985752560949, −15.352574281756787740414305251751, −13.62596227394073504074062019695, −12.53465822848824997716746600767, −11.647139875211704991151913242704, −10.93614319939657073018527048395, −9.843146428155982696323610133484, −8.91802214641144724625960682139, −8.16542158855448447263649646943, −6.543171858925352657045121368603, −5.85582449783903192534817642444, −4.919153466983595250563777648266, −3.01639448038303343184090965470, −1.51876132597866513190399591966, −0.63114450444946619071710485211, 1.09593720648970727789014245202, 1.938009513634569521271537607630, 3.763154165942292378462161115006, 5.32560235758411967417514665119, 6.467299506478768289034470391703, 6.983335360341502430044435116404, 8.072701236674001274022481375051, 9.56629578453340206357505422628, 10.579509966184468261213047190909, 10.70360787038104523469159910426, 11.97812704769027593636278194901, 13.08943175785547970379776947453, 14.42660706645037872627442618211, 15.301813542701793591253474316206, 16.65212648430057847418612890541, 17.05841909712155972094826641508, 17.960957028822274325818368308205, 18.5141239011620962236647306107, 19.53045255847156906337174496091, 20.93606970159771826839552501498, 21.25466444023285590590781136584, 22.87770659183374823101488856310, 23.26009992421294527647227954172, 24.36157561930511649028755696445, 25.45712741730670603527655096991

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