Properties

Degree 1
Conductor $ 3 \cdot 97 $
Sign $-0.994 - 0.101i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.555 − 0.831i)5-s + (0.555 − 0.831i)7-s + (−0.923 + 0.382i)8-s + (−0.980 + 0.195i)10-s + (0.923 − 0.382i)11-s + (0.831 − 0.555i)13-s + (−0.555 − 0.831i)14-s + i·16-s + (−0.831 + 0.555i)17-s + (−0.555 − 0.831i)19-s + (−0.195 + 0.980i)20-s i·22-s + (−0.980 − 0.195i)23-s + ⋯
L(s,χ)  = 1  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.555 − 0.831i)5-s + (0.555 − 0.831i)7-s + (−0.923 + 0.382i)8-s + (−0.980 + 0.195i)10-s + (0.923 − 0.382i)11-s + (0.831 − 0.555i)13-s + (−0.555 − 0.831i)14-s + i·16-s + (−0.831 + 0.555i)17-s + (−0.555 − 0.831i)19-s + (−0.195 + 0.980i)20-s i·22-s + (−0.980 − 0.195i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.994 - 0.101i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.994 - 0.101i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(291\)    =    \(3 \cdot 97\)
\( \varepsilon \)  =  $-0.994 - 0.101i$
motivic weight  =  \(0\)
character  :  $\chi_{291} (143, \cdot )$
Sato-Tate  :  $\mu(32)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 291,\ (0:\ ),\ -0.994 - 0.101i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.06058987020 - 1.189268905i$
$L(\frac12,\chi)$  $\approx$  $0.06058987020 - 1.189268905i$
$L(\chi,1)$  $\approx$  0.7044633780 - 0.8436033963i
$L(1,\chi)$  $\approx$  0.7044633780 - 0.8436033963i

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

−25.83210528019108047044073618171, −24.950066983887870906131596141326, −24.18221348990627219902065821986, −23.19847316887246288276457339050, −22.45772742225759995714536148982, −21.762907183101898354554046956048, −20.72489342694322720885510383438, −19.34224777810381454494962387414, −18.33977112525666800720608342888, −17.85048680883867206495406579389, −16.53612756840966858093934064306, −15.70725784352714584637743499187, −14.76837466590518617111211946989, −14.352999269372978245302497070825, −13.10901036690627184564791056720, −11.84786933015693698438732223357, −11.33800176929515597431731735567, −9.63178573977792073407734745214, −8.6139153299034696337731388683, −7.73573057649481907315801592971, −6.6156804118996905031725552705, −5.92342179744402737819528066678, −4.4405258849113345758363251474, −3.68079003411126130707938882124, −2.14661864966029874757718858824, 0.73095523676190624059692063292, 1.8374985722566853703968263682, 3.674337551056700158611860024708, 4.195151936288585559132380964065, 5.30660095081553321141258911837, 6.63808753147614450365508598442, 8.30122899834647147837622828940, 8.840584530148960468841025271926, 10.23558137143241080349378064087, 11.17028859411393390395349958730, 11.827674601471365819916646550719, 13.028754984347466190176867217, 13.60560408819807073306697179803, 14.732442523829612658123759856155, 15.730177721564445496680895799040, 17.01156934768828127486868401280, 17.74266194418261731466311473006, 19.01260202243464941980259472668, 20.002331345975052249071334721385, 20.27012955177473202534661534844, 21.332852438851502758631690411945, 22.25363669226271289168696603848, 23.28540678643959219994040887869, 23.960130209053931557088588048073, 24.6268140513540304917732266880

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