Properties

Label 1-171-171.31-r1-0-0
Degree $1$
Conductor $171$
Sign $-0.997 - 0.0765i$
Analytic cond. $18.3765$
Root an. cond. $18.3765$
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 + 5-s + (−0.5 + 0.866i)7-s − 8-s + (0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)20-s − 22-s + (−0.5 + 0.866i)23-s + 25-s + 26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s + (−0.5 + 0.866i)7-s − 8-s + (0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)20-s − 22-s + (−0.5 + 0.866i)23-s + 25-s + 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.997 - 0.0765i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ -0.997 - 0.0765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06709699604 + 1.750094286i\)
\(L(\frac12)\) \(\approx\) \(-0.06709699604 + 1.750094286i\)
\(L(1)\) \(\approx\) \(0.8944334228 + 0.9033628432i\)
\(L(1)\) \(\approx\) \(0.8944334228 + 0.9033628432i\)

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 \( 1 + T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
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

−26.69145787367010648056913163486, −26.12888796445927024285554111538, −24.60776217322529910249137443956, −23.81202229448080525426729016540, −22.69787691687973600256099780546, −21.932202118889449692800876016964, −20.89750350801099322711372162109, −20.32707483631677815951899449874, −18.97890237856827321494080246950, −18.31069581960153438732505316101, −17.00358501221861753600209570946, −15.95937785928201941434812660091, −14.35556525201916674793699482628, −13.59545020555238543831819071567, −13.093826749400466595590026307703, −11.58406917913687177805192633827, −10.61538191910235539055031300622, −9.76146953829849019276377581007, −8.73336208855129396265668248226, −6.76359616241001727820901640055, −5.79761436429870870713279796094, −4.50819009508613856055845712452, −3.24328469273615684692783923908, −1.98708455662157581241693677509, −0.50632220318080663162842049240, 2.1333093069018148031464879702, 3.47992597698358830008471598107, 5.15787981251878808921833779659, 5.85190542658668669453190244298, 6.8964999465729207764800272782, 8.295046421674184885624754346150, 9.285282959345058722789234853890, 10.40338715407853728251804832342, 12.22159079234832991829898399126, 12.9846135078315301408467235892, 13.80134637046592971199559211585, 15.18721419647709204236668368124, 15.605609783988243154885361093355, 17.025985334186635097198183212100, 17.80602849617710485237281544837, 18.59196613918984780073594026118, 20.27262269128305951832249203011, 21.35875522873052971632305819094, 22.094144338886617016713382085661, 22.91637616308359973968175471549, 24.04156943341056918160357332225, 25.143616140251543525567108596674, 25.582924720104071381934767988691, 26.355140751607763713331193918019, 27.85910590968082727935709932538

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