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Properties

Label 2-171-19.7-c1-0-5
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: $\chi_{171} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.875 + 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30199 - 0.335906i\)
\(L(\frac12)\) \(\approx\) \(1.30199 - 0.335906i\)
\(L(\frac{3}{2})\) 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

−12.65435616914855733321746547561, −11.82721118248572062434303083813, −10.72603439528761394181216488777, −9.709530677600590368108008012668, −8.128370173989520805560830389747, −7.82137486992859798254265091592, −6.02146487384392664345548551044, −5.17303939310479416255578736696, −4.03079140635165643282337794595, −1.45612952167493132635072227588, 2.56640660276347950610243771566, 3.52702508145343431196749731985, 4.96408746434452255443489704236, 6.95194580391858643449743756325, 7.40888691223807046524142213786, 8.585008781865755482982736275601, 10.30633156848426850857995880360, 11.06241651710061621332001288057, 11.70871198168229425769883712122, 12.58320390421066374102612759867

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