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Properties

Label 1-171-171.41-r0-0-0
Degree $1$
Conductor $171$
Sign $-0.984 - 0.174i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.984 - 0.174i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ -0.984 - 0.174i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02924106007 - 0.3329752334i\)
\(L(\frac12)\) \(\approx\) \(0.02924106007 - 0.3329752334i\)
\(L(1)\) \(\approx\) \(0.4448702648 - 0.2448916961i\)
\(L(1)\) \(\approx\) \(0.4448702648 - 0.2448916961i\)

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

−27.877200112286046429806384073, −26.971060435813315347857968113901, −25.90898588539807380164452993552, −25.5721928961483129278191918510, −24.30595602883236096289592905375, −23.2990999895085844582465496572, −22.240322769867694534101218009070, −21.17370137396437685107669054827, −19.80001135654054267571184129268, −19.06994396601425975509088304326, −18.174782496838869961371394055323, −17.52225201759040115733880698428, −15.91718771435779151655277130865, −15.44150095296930998609443934099, −14.515781961481607046997800991401, −12.93081573116777006355090377434, −11.656234309585363914471631816324, −10.52147637188048633718689870378, −9.87170405030291317813646227858, −8.42985339188775278681474153368, −7.598616169538492850689015761665, −6.34187797839672795375915097131, −5.49634753934160124499309789940, −3.2034892396300086777810259840, −2.17608480570911352604012899335, 0.34188590533630897728430442642, 1.96953832776289450880482690952, 3.55971129401094805822884947592, 4.90210699251156694949970119793, 6.6913659368792387639016338034, 7.67938011704964143054435491878, 8.77954270213378946215505569056, 9.73152579771511481886737694166, 10.71666756734580142132637777669, 11.93846742236795293081614599390, 12.83349361738762711314651476142, 13.92187872488635631285716772089, 15.848157006860391495414742309530, 16.28995287197748631223032993645, 17.20363028996940064900057946173, 18.32172409122714963563349416371, 19.33883070211640579345578477376, 20.30384603455322175185273625859, 20.79204305456527284485385765362, 22.01003462302297745673265184219, 23.50814091767629838482616925049, 24.23001384430288690009395627482, 25.40468882307469406612134049722, 26.31082198968887301850782631440, 27.05485965543829708038568239807

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