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Properties

Label 2-171-19.7-c3-0-7
Degree $2$
Conductor $171$
Sign $-0.288 - 0.957i$
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  + (1.45 + 2.52i)2-s + (−0.241 + 0.419i)4-s + (10.1 + 17.4i)5-s − 1.64·7-s + 21.8·8-s + (−29.4 + 50.9i)10-s − 8.92·11-s + (19.6 − 34.0i)13-s + (−2.39 − 4.14i)14-s + (33.8 + 58.5i)16-s + (3.95 + 6.84i)17-s + (−42.6 + 70.9i)19-s − 9.77·20-s + (−12.9 − 22.5i)22-s + (−55.2 + 95.7i)23-s + ⋯
L(s)  = 1  + (0.514 + 0.891i)2-s + (−0.0302 + 0.0523i)4-s + (0.903 + 1.56i)5-s − 0.0887·7-s + 0.967·8-s + (−0.930 + 1.61i)10-s − 0.244·11-s + (0.419 − 0.726i)13-s + (−0.0456 − 0.0791i)14-s + (0.528 + 0.915i)16-s + (0.0563 + 0.0976i)17-s + (−0.515 + 0.856i)19-s − 0.109·20-s + (−0.125 − 0.218i)22-s + (−0.501 + 0.868i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.288 - 0.957i$
Analytic conductor: \(10.0893\)
Root analytic conductor: \(3.17637\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :3/2),\ -0.288 - 0.957i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.63222 + 2.19705i\)
\(L(\frac12)\) \(\approx\) \(1.63222 + 2.19705i\)
\(L(\frac{5}{2})\) 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 \( 1 + (42.6 - 70.9i)T \)
good2 \( 1 + (-1.45 - 2.52i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-10.1 - 17.4i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + 1.64T + 343T^{2} \)
11 \( 1 + 8.92T + 1.33e3T^{2} \)
13 \( 1 + (-19.6 + 34.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-3.95 - 6.84i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (55.2 - 95.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-71.8 + 124. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 57.1T + 2.97e4T^{2} \)
37 \( 1 - 297.T + 5.06e4T^{2} \)
41 \( 1 + (112. + 194. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (163. + 283. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-56.2 + 97.4i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-305. + 529. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-368. - 638. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-420. + 728. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (53.2 - 92.1i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-386. - 669. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (369. + 639. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (257. + 446. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 618.T + 5.71e5T^{2} \)
89 \( 1 + (-62.3 + 107. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (349. + 606. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.10691425823830010260067581962, −11.42883590543871151825895217032, −10.40316450012201399803986231678, −9.955892046849952961847127280884, −8.081754865939062311707571231266, −7.05894533382096819218646152777, −6.15752427437959787819132434123, −5.51370703314719426412116286495, −3.64536772331161202930359791299, −2.08158222208045279979520750757, 1.20071454383631981979454390394, 2.45295767510329918899264673250, 4.25323613382788632076420033518, 5.04808308950224286868258194804, 6.46146070791523680206872270116, 8.143257611804119134967492012745, 9.093535756996221316614314868427, 10.07745622574242892543056138260, 11.21955036217177469241004836275, 12.19851932679573860905790537671

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