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Properties

Label 1-171-171.142-r0-0-0
Degree $1$
Conductor $171$
Sign $-0.776 + 0.630i$
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

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

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.776 + 0.630i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ -0.776 + 0.630i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4056147143 + 1.142428671i\)
\(L(\frac12)\) \(\approx\) \(0.4056147143 + 1.142428671i\)
\(L(1)\) \(\approx\) \(0.9237987763 + 0.7699123411i\)
\(L(1)\) \(\approx\) \(0.9237987763 + 0.7699123411i\)

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

−27.20332250134152933710218227946, −26.54531689375594057437892695089, −24.7061482644911330897373239432, −24.24462809407148500900878146703, −22.94772520159237251897987709610, −22.62325562075357941370855948363, −21.35898292787286140884431856540, −20.05313766160014581947287267957, −19.79420156124103641508600111767, −18.8419700520102556605139608706, −17.16291644576111574628888751300, −16.17171870299292696259318810430, −15.08571215410476586350478852458, −14.11337490171861331494671170719, −13.05345660547944718098743719843, −12.06475343078118960372949774035, −11.29509924187911952344853192970, −10.08754659769387190475632970292, −8.993534228554318523112578323049, −7.305303641078451201862887618109, −6.36894500597902648251205266307, −4.54023541331741734301727574826, −4.0980076871210965594323188171, −2.64872494376483079544853607737, −0.78719287303618270981795863445, 2.59779622260767339876143764918, 3.691984486113238109296441226873, 4.84702170723667262790529528135, 6.21748355141422931844238931928, 7.10027705109735080807636211759, 8.25531840620120487380076477729, 9.35921244253519594141965379050, 11.200656930322979926615428591244, 12.06973676531868476162704471438, 12.834459508057895369779236743490, 14.26058096241316578483049014619, 15.15347813614349745497837043452, 15.737044926993294542751100397528, 16.88249369905264329376692811731, 17.95283823324419193559351223785, 19.3437620491574458638697025970, 20.01560279853042306858985587094, 21.6489521052805160231468186442, 22.2330991188781476373916709300, 22.99930787091151448153448557221, 24.12394998083745899716989306714, 24.86985555057925202905054546009, 25.807425815681954929297912021023, 26.87793615338539168006823175361, 27.635733417616633816534919513520

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