Properties

Label 2-171-19.12-c2-0-4
Degree $2$
Conductor $171$
Sign $0.910 - 0.412i$
Analytic cond. $4.65941$
Root an. cond. $2.15856$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−2 − 3.46i)5-s + 5·7-s − 8.66i·8-s + (6 + 3.46i)10-s + 4·11-s + (10.5 + 6.06i)13-s + (−7.5 + 4.33i)14-s + (5.5 + 9.52i)16-s + (4 + 6.92i)17-s + 19·19-s + 3.99·20-s + (−6 + 3.46i)22-s + (−2 + 3.46i)23-s + ⋯
L(s)  = 1  + (−0.750 + 0.433i)2-s + (−0.125 + 0.216i)4-s + (−0.400 − 0.692i)5-s + 0.714·7-s − 1.08i·8-s + (0.600 + 0.346i)10-s + 0.363·11-s + (0.807 + 0.466i)13-s + (−0.535 + 0.309i)14-s + (0.343 + 0.595i)16-s + (0.235 + 0.407i)17-s + 19-s + 0.199·20-s + (−0.272 + 0.157i)22-s + (−0.0869 + 0.150i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ 0.910 - 0.412i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.966479 + 0.208882i\)
\(L(\frac12)\) \(\approx\) \(0.966479 + 0.208882i\)
\(L(2)\) 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 \( 1 - 19T \)
good2 \( 1 + (1.5 - 0.866i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (2 + 3.46i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 5T + 49T^{2} \)
11 \( 1 - 4T + 121T^{2} \)
13 \( 1 + (-10.5 - 6.06i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-4 - 6.92i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-48 - 27.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 5.19iT - 961T^{2} \)
37 \( 1 + 15.5iT - 1.36e3T^{2} \)
41 \( 1 + (-12 + 6.92i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-28 + 48.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (42 + 24.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (42 - 24.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-39.5 + 68.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.5 + 9.52i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (96 - 55.4i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-28.5 + 16.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 112T + 6.88e3T^{2} \)
89 \( 1 + (-66 - 38.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-96 + 55.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.44291915001688594395832842498, −11.71159641362261434318355240137, −10.43905137820371939166026015743, −9.178905334659953993992449511860, −8.497362154712958043535128993308, −7.71651997191420605503174514990, −6.50474331506494351935680482353, −4.87582581622350282015302547119, −3.69068555548219883249910035557, −1.11150846709310020999855785921, 1.15898111851617525717697489685, 2.98966419844780302698148047961, 4.72887404246960375159461306134, 6.08275203646606437763947635605, 7.57552116956633540306848179266, 8.430065414635323508633932477008, 9.534789609545284517687709110933, 10.53448663545078652314528751172, 11.26266995199753655785954059922, 11.99808360676178173231373327820

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