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Properties

Label 2-171-19.11-c1-0-4
Degree $2$
Conductor $171$
Sign $0.875 - 0.483i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.370 + 0.642i)2-s + (0.724 + 1.25i)4-s + (1.65 − 2.85i)5-s + 1.44·7-s − 2.55·8-s + (1.22 + 2.12i)10-s + 1.81·11-s + (0.5 + 0.866i)13-s + (−0.537 + 0.931i)14-s + (−0.499 + 0.866i)16-s + (−3.30 + 5.71i)17-s + (1 − 4.24i)19-s + 4.78·20-s + (−0.674 + 1.16i)22-s + (2.39 + 4.14i)23-s + ⋯
L(s)  = 1  + (−0.262 + 0.454i)2-s + (0.362 + 0.627i)4-s + (0.738 − 1.27i)5-s + 0.547·7-s − 0.904·8-s + (0.387 + 0.670i)10-s + 0.547·11-s + (0.138 + 0.240i)13-s + (−0.143 + 0.248i)14-s + (−0.124 + 0.216i)16-s + (−0.800 + 1.38i)17-s + (0.229 − 0.973i)19-s + 1.07·20-s + (−0.143 + 0.248i)22-s + (0.498 + 0.864i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.875 - 0.483i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.875 - 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19409 + 0.308067i\)
\(L(\frac12)\) \(\approx\) \(1.19409 + 0.308067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (-1 + 4.24i)T \)
good2 \( 1 + (0.370 - 0.642i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.65 + 2.85i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.44T + 7T^{2} \)
11 \( 1 - 1.81T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.30 - 5.71i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.39 - 4.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.78 + 8.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.55T + 31T^{2} \)
37 \( 1 + 5.89T + 37T^{2} \)
41 \( 1 + (1.48 - 2.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.17 + 7.22i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.48 - 2.57i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.65 + 2.85i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.21 - 7.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.17 - 12.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.78 + 8.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.17 + 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.63T + 83T^{2} \)
89 \( 1 + (-8.25 - 14.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.44 - 11.1i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.92941540116239980597959495710, −11.92156717309070714521305227565, −11.00542243849056830576402305384, −9.323986813993517024613524527837, −8.832629028797874100892133635096, −7.81434876985752383896705223528, −6.53121004477192396556870709719, −5.40587237188096489899156098875, −4.01353707060061123016859263813, −1.87374757507543874498583370423, 1.85505864880742908792514435663, 3.13451600143324871986974396707, 5.24330761978999811517949769637, 6.41154737491163570596525498977, 7.21465936273925763301072800598, 8.971888544030087968661550512881, 9.846355497656934285992104771347, 10.87373539783600939498392576555, 11.20721183171400175481411858604, 12.48770230757098761678731620228

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