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Properties

Label 2-171-19.7-c3-0-6
Degree $2$
Conductor $171$
Sign $0.373 + 0.927i$
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  + (−1.71 − 2.97i)2-s + (−1.90 + 3.29i)4-s + (6.83 + 11.8i)5-s − 20.1·7-s − 14.4·8-s + (23.4 − 40.6i)10-s + 48.0·11-s + (34.8 − 60.3i)13-s + (34.6 + 59.9i)14-s + (39.9 + 69.2i)16-s + (58.9 + 102. i)17-s + (36.1 − 74.5i)19-s − 52.0·20-s + (−82.5 − 142. i)22-s + (35.0 − 60.7i)23-s + ⋯
L(s)  = 1  + (−0.607 − 1.05i)2-s + (−0.237 + 0.412i)4-s + (0.611 + 1.05i)5-s − 1.08·7-s − 0.636·8-s + (0.743 − 1.28i)10-s + 1.31·11-s + (0.743 − 1.28i)13-s + (0.660 + 1.14i)14-s + (0.624 + 1.08i)16-s + (0.840 + 1.45i)17-s + (0.435 − 0.899i)19-s − 0.582·20-s + (−0.799 − 1.38i)22-s + (0.317 − 0.550i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.373 + 0.927i$
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: \((2,\ 171,\ (\ :3/2),\ 0.373 + 0.927i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.10321 - 0.744744i\)
\(L(\frac12)\) \(\approx\) \(1.10321 - 0.744744i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-36.1 + 74.5i)T \)
good2 \( 1 + (1.71 + 2.97i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-6.83 - 11.8i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + 20.1T + 343T^{2} \)
11 \( 1 - 48.0T + 1.33e3T^{2} \)
13 \( 1 + (-34.8 + 60.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-58.9 - 102. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-35.0 + 60.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-102. + 177. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 264.T + 2.97e4T^{2} \)
37 \( 1 + 385.T + 5.06e4T^{2} \)
41 \( 1 + (-60.4 - 104. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-156. - 270. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-57.5 + 99.6i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (186. - 323. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-131. - 228. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-143. + 248. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-240. + 417. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (17.3 + 30.0i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-352. - 610. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (293. + 509. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 898.T + 5.71e5T^{2} \)
89 \( 1 + (100. - 174. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (7.68 + 13.3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.95352393208228645805676076116, −10.79228580432226042894047061703, −10.22467828627278844319932815107, −9.504729488071588750594562274163, −8.346412099360877292903494498130, −6.51468005086190392107881549210, −6.08868210405000905902987437159, −3.54833697203810099584164953864, −2.73974073881481828238017536947, −1.01221390170544198528583559843, 1.11955468102760875098848593742, 3.50024861080384308136876453922, 5.26526533291125369241331732629, 6.38237257503174641297967983373, 7.07573661852015800192471995367, 8.624424093846576690215913970304, 9.237913466832456475375901674610, 9.814285990183516911243412009448, 11.81785963745054157668606950066, 12.35061083036666766887120432125

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