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Properties

Label 1-171-171.47-r1-0-0
Degree $1$
Conductor $171$
Sign $-0.630 - 0.776i$
Analytic cond. $18.3765$
Root an. cond. $18.3765$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.630 - 0.776i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ -0.630 - 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3613506417 - 0.7591963555i\)
\(L(\frac12)\) \(\approx\) \(0.3613506417 - 0.7591963555i\)
\(L(1)\) \(\approx\) \(0.6723072100 - 0.2740472542i\)
\(L(1)\) \(\approx\) \(0.6723072100 - 0.2740472542i\)

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

−27.41957558219104207744036696841, −26.39121523183506137086401888635, −25.92900966045967636562401837217, −25.02110934592099560033448065703, −23.84580382359343163982932353344, −23.13625632595244686932699958410, −21.85087205076870690141347192024, −20.728369318320806158732086785843, −19.60630459754701367062388631181, −18.711893724497048535609270706051, −17.68447631391295748055673607291, −16.93864452814372367014415816197, −16.05406939759920117759785538755, −14.755699693571874149812309483793, −13.94087819233832097592813698911, −12.89475553719527735986274059032, −11.08540038222126620860901689719, −9.99935962675803743088424235057, −9.65087098832953150928034308530, −7.97931254416799474352985988549, −7.08170460364159609673987061292, −6.02546726232143865586067990791, −4.912800029250370187735429066280, −2.862925958477529039235068392190, −1.3466824104820026393186669507, 0.396975303885829683668457263210, 2.17587663767512543091489538154, 2.92417638126353071410238239343, 4.90465109901804242765614234228, 6.12911765463617188905546768752, 7.648030454883401707560083057907, 8.74572502026124305849649012966, 9.802379780142815597734318817106, 10.34374886912864900269308810618, 11.99039592710198225829989767719, 12.65354443242431836791246946638, 13.666144974696833005573373799214, 15.25433249080598543190157952454, 16.4015149966919620923820780709, 17.249103118628271102711591395334, 18.32261586953441289509775356682, 18.91606324188774865751877796064, 20.21183313425955625146234146465, 21.06951143164589493275974208950, 21.80851201123757018205807001528, 22.75144698852663995857729910555, 24.49022593680816381419306875093, 25.22894628035446409043354593298, 26.014703343670092449237777812940, 26.989513939119701669748462289119

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