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Properties

Label 1-171-171.25-r0-0-0
Degree $1$
Conductor $171$
Sign $0.999 + 0.0151i$
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  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (0.173 − 0.984i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)10-s + 11-s + (0.173 + 0.984i)13-s + (0.173 − 0.984i)14-s + (0.173 − 0.984i)16-s + (0.173 − 0.984i)17-s + (−0.5 − 0.866i)20-s + (−0.939 + 0.342i)22-s + (0.766 − 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (0.173 − 0.984i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)10-s + 11-s + (0.173 + 0.984i)13-s + (0.173 − 0.984i)14-s + (0.173 − 0.984i)16-s + (0.173 − 0.984i)17-s + (−0.5 − 0.866i)20-s + (−0.939 + 0.342i)22-s + (0.766 − 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.999 + 0.0151i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.999 + 0.0151i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7796658191 + 0.005907220241i\)
\(L(\frac12)\) \(\approx\) \(0.7796658191 + 0.005907220241i\)
\(L(1)\) \(\approx\) \(0.7601148984 + 0.02455369966i\)
\(L(1)\) \(\approx\) \(0.7601148984 + 0.02455369966i\)

Euler product

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

−27.25327091519150407444328558597, −26.81958679938170759264549180345, −25.62979674176406072907514753206, −25.23829045963280466909967881900, −23.646607468475816795307038895, −22.529128230337094290998382808796, −21.71949797923948856999088736235, −20.47284500043508923290422029580, −19.54112489861210516747871555858, −18.92809813819840907700395891747, −17.57336408024871636387026467155, −17.16249341544741960021522597055, −15.82517262933629183715099918000, −14.78689252897848719069677116020, −13.506449691466053450330422256029, −12.30993048643698734847586258482, −11.00758993676533375424224237314, −10.376104493926236829466743352905, −9.42868704236229854229122194870, −8.04010516095160488551190695073, −6.98684372461979106530967373533, −6.170174618673708936991094245978, −3.81570589055992012520603941646, −2.93257505653302276488592196032, −1.23655743758394620209264849404, 1.12572700244966218104529597920, 2.565789608365617302232932738460, 4.60564161002989023755466573112, 5.93318165319324114506473115738, 6.83915823269634202055717520810, 8.39818729902335876323605911709, 9.143467630792617618680673174827, 9.80241312950069587786086828798, 11.54022023471352219931958659839, 12.166844363738186825709307895452, 13.68143451657801055122311324038, 14.90224728980865137762557435284, 16.07582708185322639112637526532, 16.619241994571906665345610444263, 17.63937487480076232064153684266, 18.80120760241260834757029501024, 19.50510467782345812111618259503, 20.590237002575397667162607133155, 21.48708758932179743943574773635, 22.859642543534984811047583230166, 24.074056338705353337411284284910, 24.9949529556879625614659429057, 25.31847994480217987860858688885, 26.694311181543543263910908628730, 27.53214770967723117580514973783

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