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Properties

Label 2-171-19.11-c3-0-6
Degree $2$
Conductor $171$
Sign $-0.824 - 0.565i$
Analytic cond. $10.0893$
Root an. cond. $3.17637$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.72 + 4.72i)2-s + (−10.8 − 18.8i)4-s + (−1.48 + 2.56i)5-s + 6.79·7-s + 74.7·8-s + (−8.07 − 13.9i)10-s + 51.5·11-s + (−10.0 − 17.3i)13-s + (−18.5 + 32.0i)14-s + (−116. + 202. i)16-s + (−53.3 + 92.4i)17-s + (80.5 − 19.1i)19-s + 64.2·20-s + (−140. + 243. i)22-s + (−29.6 − 51.3i)23-s + ⋯
L(s)  = 1  + (−0.963 + 1.66i)2-s + (−1.35 − 2.35i)4-s + (−0.132 + 0.229i)5-s + 0.366·7-s + 3.30·8-s + (−0.255 − 0.442i)10-s + 1.41·11-s + (−0.213 − 0.369i)13-s + (−0.353 + 0.612i)14-s + (−1.82 + 3.16i)16-s + (−0.761 + 1.31i)17-s + (0.972 − 0.231i)19-s + 0.718·20-s + (−1.36 + 2.35i)22-s + (−0.268 − 0.465i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 - 0.565i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.824 - 0.565i$
Analytic conductor: \(10.0893\)
Root analytic conductor: \(3.17637\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :3/2),\ -0.824 - 0.565i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.275739 + 0.889781i\)
\(L(\frac12)\) \(\approx\) \(0.275739 + 0.889781i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-80.5 + 19.1i)T \)
good2 \( 1 + (2.72 - 4.72i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (1.48 - 2.56i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 - 6.79T + 343T^{2} \)
11 \( 1 - 51.5T + 1.33e3T^{2} \)
13 \( 1 + (10.0 + 17.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (53.3 - 92.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (29.6 + 51.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-39.6 - 68.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 158.T + 2.97e4T^{2} \)
37 \( 1 - 244.T + 5.06e4T^{2} \)
41 \( 1 + (172. - 298. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-168. + 291. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-114. - 198. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-324. - 562. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-46.5 + 80.6i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-297. - 514. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-494. - 857. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (420. - 729. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-439. + 760. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (172. - 299. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 145.T + 5.71e5T^{2} \)
89 \( 1 + (561. + 973. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-62.5 + 108. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.02230818236959028863887278794, −11.38120015346113178654138420598, −10.34558571471028745864139272269, −9.252506327715293232665314830113, −8.549483289459551790365261182171, −7.45428772276256967329591367719, −6.61893041788724730417752584685, −5.61158661656099611201261473702, −4.25221517515853202054770017726, −1.24258060071307823470611880751, 0.72880509463882701710982613474, 2.09703875418434453769281335494, 3.60800363650865938269405887142, 4.71479898712054053284022368080, 7.04771302046311586800898724751, 8.228236959705705733281956470111, 9.248898085270533018672133830126, 9.723807257869595596722821008500, 11.13630019789508622730284286273, 11.66569806200015461255229049360

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