Properties

Degree 1
Conductor 97
Sign $0.947 + 0.320i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.130 − 0.991i)2-s + (−0.608 − 0.793i)3-s + (−0.965 − 0.258i)4-s + (−0.946 − 0.321i)5-s + (−0.866 + 0.5i)6-s + (0.751 + 0.659i)7-s + (−0.382 + 0.923i)8-s + (−0.258 + 0.965i)9-s + (−0.442 + 0.896i)10-s + (−0.991 + 0.130i)11-s + (0.382 + 0.923i)12-s + (0.321 − 0.946i)13-s + (0.751 − 0.659i)14-s + (0.321 + 0.946i)15-s + (0.866 + 0.5i)16-s + (0.659 + 0.751i)17-s + ⋯
L(s,χ)  = 1  + (0.130 − 0.991i)2-s + (−0.608 − 0.793i)3-s + (−0.965 − 0.258i)4-s + (−0.946 − 0.321i)5-s + (−0.866 + 0.5i)6-s + (0.751 + 0.659i)7-s + (−0.382 + 0.923i)8-s + (−0.258 + 0.965i)9-s + (−0.442 + 0.896i)10-s + (−0.991 + 0.130i)11-s + (0.382 + 0.923i)12-s + (0.321 − 0.946i)13-s + (0.751 − 0.659i)14-s + (0.321 + 0.946i)15-s + (0.866 + 0.5i)16-s + (0.659 + 0.751i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.947 + 0.320i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.947 + 0.320i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $0.947 + 0.320i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (76, \cdot )$
Sato-Tate  :  $\mu(96)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 97,\ (1:\ ),\ 0.947 + 0.320i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4312926023 + 0.07092173387i$
$L(\frac12,\chi)$  $\approx$  $0.4312926023 + 0.07092173387i$
$L(\chi,1)$  $\approx$  0.5278904103 - 0.3471297493i
$L(1,\chi)$  $\approx$  0.5278904103 - 0.3471297493i

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

−29.912475612261187697731814364955, −28.33471867211230178087178311756, −27.3674005436260199502133760057, −26.72863600637502662136309859990, −25.95754463464224596872186204154, −24.207680756689450783672266030634, −23.40055404544315094868724845882, −22.91297563223167781258550132634, −21.5244868447509143449918343349, −20.60418806478095086356405924191, −18.77813237529109907545670909777, −17.85890995409633717313646924842, −16.50868953083326820168918386251, −16.02803588118973964560452985014, −14.84243249155001103517349426601, −13.96953883900046380413184741978, −12.191909197031565630603496962664, −11.06641444464082201343680996732, −9.85446151962359129241735838925, −8.25697286055330421541311559528, −7.31722634129365948638288236039, −5.837131202152510462943322117323, −4.55541230618800458353401005, −3.729387092546796125488648700472, −0.22899888934307631735437797184, 1.28874745598585977103595170019, 2.858588230376336673651147141468, 4.70408748084294776609337870002, 5.653654197878378495206702500691, 7.7954974139198776200229683523, 8.50966778027091733881796326707, 10.53955788742438636001822292400, 11.36361725368271978258485722579, 12.41550543725545316794740383278, 12.997836184111320621443522953173, 14.59046492989233749171737977583, 15.91654422335561451436308547881, 17.61285961458782445988850924840, 18.27465531508234146404233236315, 19.29821505468100282498940309522, 20.26890921410413686356002323258, 21.46319200998194095513436073353, 22.5891518296732342581381493774, 23.67827296072419498633804152929, 24.082319640505446365059118966947, 25.75843239318444998396634155142, 27.4432363188510479109474609464, 27.982809924129612375529430722177, 28.74282813597497212021578061864, 30.05229272925444675456533371226

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