Properties

Degree 1
Conductor $ 3 \cdot 31 $
Sign $0.644 + 0.764i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.5 − 0.866i)5-s + (−0.104 + 0.994i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)10-s + (0.913 − 0.406i)11-s + (0.978 + 0.207i)13-s + (−0.913 − 0.406i)14-s + (0.309 + 0.951i)16-s + (0.913 + 0.406i)17-s + (−0.978 + 0.207i)19-s + (−0.913 + 0.406i)20-s + (0.104 + 0.994i)22-s + (−0.809 + 0.587i)23-s + ⋯
L(s,χ)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.5 − 0.866i)5-s + (−0.104 + 0.994i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)10-s + (0.913 − 0.406i)11-s + (0.978 + 0.207i)13-s + (−0.913 − 0.406i)14-s + (0.309 + 0.951i)16-s + (0.913 + 0.406i)17-s + (−0.978 + 0.207i)19-s + (−0.913 + 0.406i)20-s + (0.104 + 0.994i)22-s + (−0.809 + 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.644 + 0.764i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.644 + 0.764i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(93\)    =    \(3 \cdot 31\)
\( \varepsilon \)  =  $0.644 + 0.764i$
motivic weight  =  \(0\)
character  :  $\chi_{93} (44, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 93,\ (0:\ ),\ 0.644 + 0.764i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8177470731 + 0.3801607914i$
$L(\frac12,\chi)$  $\approx$  $0.8177470731 + 0.3801607914i$
$L(\chi,1)$  $\approx$  0.8849323106 + 0.3191996883i
$L(1,\chi)$  $\approx$  0.8849323106 + 0.3191996883i

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

−30.116005290283912241609278559087, −29.3943156370061241349051261012, −28.05660058778913872897309409113, −27.1840764377145755042654380201, −26.08540874664364572445482547434, −25.40772392167307546696271083082, −23.37357559301347061675434257774, −22.67177221606443943062097316788, −21.58798581047182147741873553821, −20.535343551349940145639319207390, −19.563229397300104912483642973158, −18.44360726739691168401688590641, −17.54514975825138872132767284622, −16.50600464800180505590917464785, −14.50861022434731949222093416663, −13.75395938388484376375535827892, −12.50288514747104650837582756205, −11.07103234401201490557506212230, −10.35157981402288467439674478609, −9.25286371998944281050767408457, −7.6900922197745420583371300563, −6.3346705623567714152154786536, −4.28161389829273158322913359846, −3.136349303167856520592681707384, −1.47931506927639906725195393775, 1.52517226503800344600175530956, 4.0750194780106983438059818027, 5.65282994898765282612186143140, 6.27467569483574455046850893789, 8.24256345662485151465873857281, 8.91289269560341479301293854878, 10.01368763335637373072316415574, 11.894568969909892310811911870527, 13.177279977323308502739260458234, 14.249196794666479502773095774155, 15.48313019883366573843221843868, 16.49325337625473610192029919884, 17.33694950976199055620342927145, 18.54633105827888456499775103699, 19.48157266622491133306784243266, 21.1057868148072763534427632556, 22.04259587737108536816589606513, 23.3994779123827629017903519765, 24.373388335989287272204135223179, 25.29123051757172409392116087487, 25.84560359968938992432912008334, 27.59326994350795334185365140861, 27.98381496821786321735968927050, 29.09881731006108173953144707672, 30.62304636266016524860946067023

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