Properties

Degree 1
Conductor $ 3 \cdot 17 $
Sign $0.238 - 0.971i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.707 − 0.707i)2-s i·4-s + (−0.923 − 0.382i)5-s + (0.923 − 0.382i)7-s + (−0.707 − 0.707i)8-s + (−0.923 + 0.382i)10-s + (0.382 + 0.923i)11-s + i·13-s + (0.382 − 0.923i)14-s − 16-s + (−0.707 + 0.707i)19-s + (−0.382 + 0.923i)20-s + (0.923 + 0.382i)22-s + (−0.382 − 0.923i)23-s + (0.707 + 0.707i)25-s + (0.707 + 0.707i)26-s + ⋯
L(s,χ)  = 1  + (0.707 − 0.707i)2-s i·4-s + (−0.923 − 0.382i)5-s + (0.923 − 0.382i)7-s + (−0.707 − 0.707i)8-s + (−0.923 + 0.382i)10-s + (0.382 + 0.923i)11-s + i·13-s + (0.382 − 0.923i)14-s − 16-s + (−0.707 + 0.707i)19-s + (−0.382 + 0.923i)20-s + (0.923 + 0.382i)22-s + (−0.382 − 0.923i)23-s + (0.707 + 0.707i)25-s + (0.707 + 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.238 - 0.971i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.238 - 0.971i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(51\)    =    \(3 \cdot 17\)
\( \varepsilon \)  =  $0.238 - 0.971i$
motivic weight  =  \(0\)
character  :  $\chi_{51} (29, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 51,\ (0:\ ),\ 0.238 - 0.971i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8768714302 - 0.6878087264i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8768714302 - 0.6878087264i\)
\(L(\chi,1)\)  \(\approx\)  \(1.118990202 - 0.5837684149i\)
\(L(1,\chi)\)  \(\approx\)  \(1.118990202 - 0.5837684149i\)

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

−34.05668233690729211704622671126, −32.55638429337146369595363718120, −31.63696592758724393532120355758, −30.59598576185048578159877833719, −29.81646027339673789367472043723, −27.67420040002063803342042641133, −26.98226675869364566545087725780, −25.59646449358668565302656868587, −24.36018381542398119910349520551, −23.589319830984523924372716261376, −22.337384578744002623733749669340, −21.35882573092026618023471689979, −19.88235793608283628993163048328, −18.32568164680878587110272435162, −17.06742768106494846069937412094, −15.59433341135039939479375539737, −14.90015998754734215698299983656, −13.58267559453329034761973112392, −12.02065205729556320135025716498, −11.08628481929251637483794795949, −8.54642394899521866228408359189, −7.66051660214625191240559855792, −6.05057675117467215162903052328, −4.5450902993373514860405075696, −3.0699494678945929150875363014, 1.739075863166180118857646866047, 3.97411303218697057953674487212, 4.816039720213551788389687218605, 6.90607991110009930244885097899, 8.64647249886642141240674435188, 10.39413121888865520193651360809, 11.6708507982240889872193283378, 12.48109068454064541464935859625, 14.15756584758768529696537631460, 15.042304834721314079305021461270, 16.58118517426058477763562360704, 18.29351294486290591905034110516, 19.63673506348639889275346982303, 20.47347228316653242574791505914, 21.55731221663168487421268690902, 23.09677060800308937228171438760, 23.72139539882593082108053261087, 24.8832508553519599609458866532, 26.89385281160561882722561808281, 27.806860627913248983046487428641, 28.77580912892054948938194918995, 30.32308456863517922869794418859, 30.91870230928163620393722838814, 31.94608154075487388312663014399, 33.19648132819587302070410513530

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