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Properties

Label 1-171-171.29-r0-0-0
Degree $1$
Conductor $171$
Sign $0.150 + 0.988i$
Analytic cond. $0.794120$
Root an. cond. $0.794120$
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  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 − 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (0.939 + 0.342i)10-s − 11-s + (0.939 + 0.342i)13-s + (−0.939 + 0.342i)14-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + (0.5 + 0.866i)20-s + (−0.766 − 0.642i)22-s + (−0.173 − 0.984i)23-s + (0.766 − 0.642i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 − 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (0.939 + 0.342i)10-s − 11-s + (0.939 + 0.342i)13-s + (−0.939 + 0.342i)14-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + (0.5 + 0.866i)20-s + (−0.766 − 0.642i)22-s + (−0.173 − 0.984i)23-s + (0.766 − 0.642i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.150 + 0.988i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.150 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.361308262 + 1.169409967i\)
\(L(\frac12)\) \(\approx\) \(1.361308262 + 1.169409967i\)
\(L(1)\) \(\approx\) \(1.430596886 + 0.7447848116i\)
\(L(1)\) \(\approx\) \(1.430596886 + 0.7447848116i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−27.56513839612575350580154271454, −26.11081282554117027121924511258, −25.528657295311343555894183129931, −24.15314436558643146202777288856, −23.19548185350988709689737320477, −22.599362124900766957929077409128, −21.245506034830754390686392232378, −20.90824759057037050185248848208, −19.68198707224070610673434553481, −18.6625068554979777334262381048, −17.70246202479314533669294917641, −16.33097076640077593720137194002, −15.24406786381939354584526036250, −13.96821120212957359605139682314, −13.42898486882796956791613667638, −12.55162084345237660421461612884, −10.95345541151680741310748102108, −10.35304560406141683477951043703, −9.43332410445623954568260584603, −7.549590757169261466638757195200, −6.182574435886192619517531935071, −5.41914108357699140045757911964, −3.83668162396154699789768955913, −2.82278099746775955173979098941, −1.352065340017627580519994626844, 2.18315388504168541907581788714, 3.36963221266494358786976626268, 5.07901538756642446301953384113, 5.750805211896989291465934381619, 6.78431427859976657198456261828, 8.29443752364934360795177333573, 9.18326558002738361068014656719, 10.600337398170038810134718012554, 12.15176323987518304240908479556, 12.906879112335681390443548508088, 13.785683848057154457584008603677, 14.80668925215303703936435377208, 16.05355532209385931300900563878, 16.51368044461708523603022899100, 17.96318916095980444677824637684, 18.60576434735928849407917150463, 20.470014877489922248279956317807, 21.18861727552884575469960928484, 21.944317974368953876761600465785, 22.97758724629830929640346617223, 23.93947541223138202817383011014, 24.93816144966442898858696440268, 25.680112295411709032212778052007, 26.238064119841339883298919436702, 27.88108048027349569909561325052

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