Properties

Degree 1
Conductor 151
Sign $-0.877 - 0.480i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.978 − 0.207i)2-s + (0.0627 − 0.998i)3-s + (0.913 + 0.406i)4-s + (0.604 − 0.796i)5-s + (−0.268 + 0.963i)6-s + (−0.570 − 0.821i)7-s + (−0.809 − 0.587i)8-s + (−0.992 − 0.125i)9-s + (−0.756 + 0.653i)10-s + (−0.895 − 0.444i)11-s + (0.463 − 0.886i)12-s + (0.985 + 0.166i)13-s + (0.387 + 0.921i)14-s + (−0.756 − 0.653i)15-s + (0.669 + 0.743i)16-s + (0.996 + 0.0836i)17-s + ⋯
L(s,χ)  = 1  + (−0.978 − 0.207i)2-s + (0.0627 − 0.998i)3-s + (0.913 + 0.406i)4-s + (0.604 − 0.796i)5-s + (−0.268 + 0.963i)6-s + (−0.570 − 0.821i)7-s + (−0.809 − 0.587i)8-s + (−0.992 − 0.125i)9-s + (−0.756 + 0.653i)10-s + (−0.895 − 0.444i)11-s + (0.463 − 0.886i)12-s + (0.985 + 0.166i)13-s + (0.387 + 0.921i)14-s + (−0.756 − 0.653i)15-s + (0.669 + 0.743i)16-s + (0.996 + 0.0836i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.877 - 0.480i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.877 - 0.480i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(151\)
\( \varepsilon \)  =  $-0.877 - 0.480i$
motivic weight  =  \(0\)
character  :  $\chi_{151} (90, \cdot )$
Sato-Tate  :  $\mu(75)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 151,\ (0:\ ),\ -0.877 - 0.480i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1601641177 - 0.6262599836i$
$L(\frac12,\chi)$  $\approx$  $0.1601641177 - 0.6262599836i$
$L(\chi,1)$  $\approx$  0.5200350268 - 0.4509690214i
$L(1,\chi)$  $\approx$  0.5200350268 - 0.4509690214i

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

−28.172362755508869632586999604003, −27.64564784314787890904395025816, −26.18674492330148277627625083319, −25.86683644087071835280642697723, −25.23861042730836586675216543290, −23.51554790757248485135760598390, −22.482448128527067575566623796652, −21.301409360704853595935014107043, −20.75263273010763059126998890988, −19.30417957702048198825531318860, −18.48456379291589517484893369078, −17.540911716133178611477873062812, −16.421322488587139545665012401204, −15.38049022846406434785420343682, −14.966128945652872834338738050160, −13.392328473486277049669074794863, −11.6395732095557723060724466784, −10.61697667385758217037898679994, −9.85033014637822069180557981151, −9.02777216112906155619766377524, −7.755348314783062852214737114835, −6.20123735194264427613312123741, −5.47969845042974107152304980508, −3.28653718631433711880735112508, −2.27908185222311739431344967781, 0.73688332783669305806512421029, 1.940691603369772421133982417992, 3.438799803012225384853792083631, 5.73668391356184486252562218680, 6.69793592249447826603481528672, 7.9870949758319947557146839491, 8.70341199174198924079966644421, 10.0432404934995689208579894132, 11.01707930400610190786010430038, 12.54122117519204290874226642904, 13.04085076018673720460717536900, 14.261715980399790747908368266422, 16.240186662823476292516784984689, 16.68037339711229106930195824355, 17.8175055014167973229060791405, 18.6542509336927980263408047093, 19.54175600177260191548816859063, 20.60026256731244211025585664223, 21.18989549158080239379412349883, 23.08770719085131644093599925492, 23.92915768113494232543699410935, 24.94080170170125965491316634233, 25.76822227632367160483404555015, 26.40321249865366823212578189764, 27.97318940632795571096999215939

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