# Properties

 Label 1-171-171.56-r0-0-0 Degree $1$ Conductor $171$ Sign $-0.642 - 0.766i$ Analytic cond. $0.794120$ Root an. cond. $0.794120$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + 8-s − 10-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 17-s + (0.5 + 0.866i)20-s + (0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯
 L(s)  = 1 + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + 8-s − 10-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 17-s + (0.5 + 0.866i)20-s + (0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$171$$    =    $$3^{2} \cdot 19$$ Sign: $-0.642 - 0.766i$ Analytic conductor: $$0.794120$$ Root analytic conductor: $$0.794120$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{171} (56, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 171,\ (0:\ ),\ -0.642 - 0.766i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.3434851566 - 0.7366062955i$$ $$L(\frac12)$$ $$\approx$$ $$0.3434851566 - 0.7366062955i$$ $$L(1)$$ $$\approx$$ $$0.6435829491 - 0.5028407543i$$ $$L(1)$$ $$\approx$$ $$0.6435829491 - 0.5028407543i$$

## Euler product

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

## Imaginary part of the first few zeros on the critical line

−27.70298865370428374446514001305, −26.65948378100613366273787297767, −25.97655910945148047941817680364, −25.13238099937775350997835443521, −24.30153674700890116323124039474, −23.13117353010819709210670334397, −22.15104467897041135322095467377, −21.47493128037702747434140886997, −19.60714138297543938607595561067, −18.86466438402608653019385884045, −18.14822511760038064078931559787, −17.09003934064236912978128224300, −16.00519951219929538949331145030, −15.17987487555606099181430158174, −14.10566213038024571540964108796, −13.35008399995231340124612927610, −11.559616424062233336173123901310, −10.54293700797572694758277804320, −9.24867571797387473097018111421, −8.69821308106273488866745740271, −6.96966954398645735624599475719, −6.35275440823146184634714725645, −5.31988209104059414073439249037, −3.4866951803111592545034787832, −1.83432332824511976978181584811, 0.82314565164290351537262736868, 2.20745619004202679562694685874, 3.81166984546190661611224856795, 4.79322486198512519635360032583, 6.563579592668694751571413618725, 7.93186788133706567440865193024, 9.05955445453418852856850947987, 9.916081108036628569264551084647, 10.83503677046741146401509581558, 12.20151874275598864995177389749, 13.07567168090784696766715747017, 13.720116176645425228273405531371, 15.49424574059538021851327638422, 16.86655033265711795697942832835, 17.280861290785885545181656331298, 18.36872087402135178370614685341, 19.73552843189557389624942920695, 20.305755573454304534456383064062, 20.9825833541462063556169903324, 22.36673972656746897368787363307, 22.96645076906060405447659941504, 24.48688472507643783944030521535, 25.47728827257489517155143121869, 26.32263007616976235176706423964, 27.39156943170221553073857280030