Properties

Degree 1
Conductor 373
Sign $0.930 - 0.367i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.890 + 0.455i)2-s + (0.857 − 0.514i)3-s + (0.585 + 0.810i)4-s + (−0.184 − 0.982i)5-s + (0.997 − 0.0675i)6-s + (0.820 − 0.571i)7-s + (0.151 + 0.988i)8-s + (0.470 − 0.882i)9-s + (0.283 − 0.959i)10-s + (−0.931 + 0.363i)11-s + (0.918 + 0.394i)12-s + (0.151 − 0.988i)13-s + (0.990 − 0.134i)14-s + (−0.664 − 0.747i)15-s + (−0.315 + 0.948i)16-s + (−0.994 + 0.101i)17-s + ⋯
L(s,χ)  = 1  + (0.890 + 0.455i)2-s + (0.857 − 0.514i)3-s + (0.585 + 0.810i)4-s + (−0.184 − 0.982i)5-s + (0.997 − 0.0675i)6-s + (0.820 − 0.571i)7-s + (0.151 + 0.988i)8-s + (0.470 − 0.882i)9-s + (0.283 − 0.959i)10-s + (−0.931 + 0.363i)11-s + (0.918 + 0.394i)12-s + (0.151 − 0.988i)13-s + (0.990 − 0.134i)14-s + (−0.664 − 0.747i)15-s + (−0.315 + 0.948i)16-s + (−0.994 + 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.930 - 0.367i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.930 - 0.367i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(373\)
\( \varepsilon \)  =  $0.930 - 0.367i$
motivic weight  =  \(0\)
character  :  $\chi_{373} (38, \cdot )$
Sato-Tate  :  $\mu(93)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 373,\ (0:\ ),\ 0.930 - 0.367i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.839852119 - 0.5405893598i$
$L(\frac12,\chi)$  $\approx$  $2.839852119 - 0.5405893598i$
$L(\chi,1)$  $\approx$  2.191828730 - 0.1557283538i
$L(1,\chi)$  $\approx$  2.191828730 - 0.1557283538i

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

−24.360819896341880367079944539379, −23.96744141056860336379577637756, −22.55316463769652405969468224269, −21.99329600332174164266523909878, −21.169385729180911221900632057422, −20.60716650576726359270432531366, −19.48979510415771246322863427732, −18.77340905381856739509643495042, −18.00510843380573462914937369162, −16.0291965295924116450526362841, −15.585301891242938231358440690867, −14.66889324750394244442259633229, −14.01745399634829912856274346520, −13.34894313578882434544924354677, −11.857912487291843688433948728312, −11.13822988811534732870567181303, −10.35565122061054491159465058604, −9.28328669562225719141596280429, −8.06424495835046418458618714151, −7.045657794466835148779543270174, −5.76416890007858459791218064855, −4.65263194554685718791711325627, −3.77265199565255289642442802078, −2.58493357315283316354033233277, −2.0860769177126267920775239913, 1.381613264093751127697491641259, 2.60434059221884242508512878539, 3.84786539214367156704210475738, 4.74698942699340658983774454586, 5.71614413952009324430513055721, 7.18607049827174346227543928649, 7.963614541080337351531988883775, 8.39455410040370011769627224865, 9.88329109744852715016122610567, 11.32813764599673340261129607538, 12.278765341992656454623053340241, 13.33710186136963820695786381797, 13.44872696191618885932004996910, 14.76424486912931787631609757132, 15.45714045758327268168249374368, 16.30552673252601400460887546164, 17.58565169080570498783486781170, 18.08449712569747198260464856357, 19.83125136374628437084033018364, 20.41129581574259357258474852273, 20.75471624755759927717734674324, 21.89824163758001823386678801701, 23.22032629237418810536166535273, 23.81293718044095895475836952300, 24.51656578145917801783502803057

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