Properties

Degree 1
Conductor 373
Sign $0.981 - 0.190i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.266 − 0.963i)2-s + (−0.0843 − 0.996i)3-s + (−0.857 − 0.514i)4-s + (0.993 + 0.117i)5-s + (−0.982 − 0.184i)6-s + (0.994 + 0.101i)7-s + (−0.724 + 0.688i)8-s + (−0.985 + 0.168i)9-s + (0.378 − 0.925i)10-s + (−0.234 + 0.972i)11-s + (−0.440 + 0.897i)12-s + (−0.688 + 0.724i)13-s + (0.363 − 0.931i)14-s + (0.0337 − 0.999i)15-s + (0.470 + 0.882i)16-s + (0.874 − 0.485i)17-s + ⋯
L(s,χ)  = 1  + (0.266 − 0.963i)2-s + (−0.0843 − 0.996i)3-s + (−0.857 − 0.514i)4-s + (0.993 + 0.117i)5-s + (−0.982 − 0.184i)6-s + (0.994 + 0.101i)7-s + (−0.724 + 0.688i)8-s + (−0.985 + 0.168i)9-s + (0.378 − 0.925i)10-s + (−0.234 + 0.972i)11-s + (−0.440 + 0.897i)12-s + (−0.688 + 0.724i)13-s + (0.363 − 0.931i)14-s + (0.0337 − 0.999i)15-s + (0.470 + 0.882i)16-s + (0.874 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.981 - 0.190i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.981 - 0.190i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(373\)
\( \varepsilon \)  =  $0.981 - 0.190i$
motivic weight  =  \(0\)
character  :  $\chi_{373} (85, \cdot )$
Sato-Tate  :  $\mu(372)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 373,\ (1:\ ),\ 0.981 - 0.190i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.762154716 - 0.1690472716i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.762154716 - 0.1690472716i\)
\(L(\chi,1)\)  \(\approx\)  \(1.052160220 - 0.5933571843i\)
\(L(1,\chi)\)  \(\approx\)  \(1.052160220 - 0.5933571843i\)

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

−24.49302502355186954190945310757, −23.66842783455245234902151208922, −22.58588107509039421773162981443, −21.726005881854215990614072840, −21.29681195140593581309637799426, −20.47767033363311224045740038476, −18.91659411043000672520201772906, −17.75682064694583448347216067266, −17.14076390740044810913791932569, −16.590046091959140952676405267665, −15.405410546560678620681897193924, −14.66258273432142660778325029282, −14.02427127700926100600754663228, −13.06055200565112656412873155025, −11.79271335358314137776855301630, −10.49007137332239492885322121723, −9.83916416795799920426592931407, −8.551858740283795605097466413314, −8.12171348765839116240730298593, −6.45850137603815821414603487500, −5.5177817701992873588238115278, −4.98404728265366840265077930436, −3.84917571263862567059598703268, −2.516571446562229950098348094678, −0.43847825905806058120677690065, 1.513532068405684921610064012233, 1.83845312915857383049058480457, 2.975605808580563862996473013847, 4.74483046841105434857230214777, 5.41290323140819098499974237739, 6.626857768924505657105259888640, 7.805318972158747230134420098132, 8.92920100247585589182494901224, 9.94817098572806794119184651675, 10.84437022997968978233244929877, 12.06714937196731535982794870159, 12.37636840053648488970187853334, 13.62763349280210316986314194171, 14.2141872678021138507689037660, 14.86939935322000440887403569123, 16.92125252547830479108356606412, 17.66859435298026572943380389992, 18.22218030216271172890540884875, 19.04109929994763236539066442567, 20.04231511685382775922442203238, 20.94985493479919254252450916318, 21.55463953735647627500821765070, 22.599479039034582472804053096789, 23.48293391752075096750558218156, 24.15815184895579947874951159508

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