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Properties

Label 1-171-171.113-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 $0$

Origins

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Normalization:  

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: $\chi_{171} (113, \cdot )$
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 \( 1 + (0.5 - 0.866i)T \)
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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

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

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