Properties

Degree 1
Conductor 163
Sign $0.0727 + 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.963 + 0.268i)2-s + (0.657 + 0.753i)3-s + (0.856 − 0.516i)4-s + (0.597 − 0.802i)5-s + (−0.835 − 0.549i)6-s + (−0.565 + 0.824i)7-s + (−0.686 + 0.727i)8-s + (−0.135 + 0.990i)9-s + (−0.360 + 0.932i)10-s + (−0.981 + 0.192i)11-s + (0.952 + 0.305i)12-s + (−0.286 + 0.957i)13-s + (0.323 − 0.946i)14-s + (0.996 − 0.0774i)15-s + (0.466 − 0.884i)16-s + (0.973 + 0.230i)17-s + ⋯
L(s,χ)  = 1  + (−0.963 + 0.268i)2-s + (0.657 + 0.753i)3-s + (0.856 − 0.516i)4-s + (0.597 − 0.802i)5-s + (−0.835 − 0.549i)6-s + (−0.565 + 0.824i)7-s + (−0.686 + 0.727i)8-s + (−0.135 + 0.990i)9-s + (−0.360 + 0.932i)10-s + (−0.981 + 0.192i)11-s + (0.952 + 0.305i)12-s + (−0.286 + 0.957i)13-s + (0.323 − 0.946i)14-s + (0.996 − 0.0774i)15-s + (0.466 − 0.884i)16-s + (0.973 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0727 + 0.997i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0727 + 0.997i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(163\)
\( \varepsilon \)  =  $0.0727 + 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{163} (14, \cdot )$
Sato-Tate  :  $\mu(81)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 163,\ (0:\ ),\ 0.0727 + 0.997i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6512078627 + 0.6054267108i$
$L(\frac12,\chi)$  $\approx$  $0.6512078627 + 0.6054267108i$
$L(\chi,1)$  $\approx$  0.7879344812 + 0.3607908479i
$L(1,\chi)$  $\approx$  0.7879344812 + 0.3607908479i

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

−27.15349162795178229845819777916, −26.45405884299282001222944331307, −25.68855977367631869808695980650, −25.0813799778534106977599003650, −23.8183331566067573067919389044, −22.67867002923123423098904078094, −21.29673017698665211745314231443, −20.35641062166449725288457706680, −19.571842497660431615535812457004, −18.509327423066392385595953582958, −18.02193993847147916794771894006, −16.91830755905704867266824955020, −15.63741349057010170893813886587, −14.43349102742540811856869434624, −13.30646740435935610844477191030, −12.47489129689864640322098176697, −10.86271056899082530156790611443, −10.11994024687881833182135348837, −9.08274690585709771666020371115, −7.5861712043652184891900047005, −7.22266573232303222460272168509, −5.882900183448163625321989608616, −3.18606907553544326432747397656, −2.70623102467497917423848084661, −0.97236631017563021377925683190, 1.83077772699294207496601300452, 3.00272880083803279288065112437, 5.0003673237834236861396946598, 5.90857452060650773094590607284, 7.63907333541824943134458950807, 8.63106925214555984335429762572, 9.609654519646047941242892789084, 9.968442126823150668125075399563, 11.57297153797323000443760911252, 12.89375438318379667908772204371, 14.233975729226408307025633248414, 15.30884725986915727316937126075, 16.234323489020539960753625260075, 16.77483849287389306853131521513, 18.21621777439821395224842973713, 19.10862813743457440916935852235, 20.0881318010385298417537263576, 21.11362830562105086966990869604, 21.57667493788054352690648043856, 23.32000595751774493153027287999, 24.61347755392109340761305218190, 25.318290305658821361587282725931, 25.970419648751046377896595605776, 26.90565483102055727799680602361, 27.99046297969411592352271069499

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