Properties

Label 1-51-51.29-r0-0-0
Degree $1$
Conductor $51$
Sign $0.238 - 0.971i$
Analytic cond. $0.236843$
Root an. cond. $0.236843$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.238 - 0.971i$
Analytic conductor: \(0.236843\)
Root analytic conductor: \(0.236843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 51,\ (0:\ ),\ 0.238 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8768714302 - 0.6878087264i\)
\(L(\frac12)\) \(\approx\) \(0.8768714302 - 0.6878087264i\)
\(L(1)\) \(\approx\) \(1.118990202 - 0.5837684149i\)
\(L(1)\) \(\approx\) \(1.118990202 - 0.5837684149i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + iT \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.382 - 0.923i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (-0.923 + 0.382i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.923 - 0.382i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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