Properties

Label 2-171-19.8-c2-0-0
Degree $2$
Conductor $171$
Sign $-0.999 - 0.0226i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.06 + 0.617i)2-s + (−1.23 − 2.14i)4-s + (−4.08 + 7.08i)5-s − 10.4·7-s − 7.99i·8-s + (−8.73 + 5.04i)10-s − 3.90·11-s + (−2.02 + 1.16i)13-s + (−11.1 − 6.46i)14-s + (−0.0227 + 0.0394i)16-s + (2.13 − 3.70i)17-s + (−17.4 + 7.45i)19-s + 20.2·20-s + (−4.17 − 2.40i)22-s + (20.0 + 34.7i)23-s + ⋯
L(s)  = 1  + (0.534 + 0.308i)2-s + (−0.309 − 0.536i)4-s + (−0.817 + 1.41i)5-s − 1.49·7-s − 0.999i·8-s + (−0.873 + 0.504i)10-s − 0.354·11-s + (−0.155 + 0.0898i)13-s + (−0.799 − 0.461i)14-s + (−0.00142 + 0.00246i)16-s + (0.125 − 0.217i)17-s + (−0.919 + 0.392i)19-s + 1.01·20-s + (−0.189 − 0.109i)22-s + (0.872 + 1.51i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0226i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.999 - 0.0226i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ -0.999 - 0.0226i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00375930 + 0.332399i\)
\(L(\frac12)\) \(\approx\) \(0.00375930 + 0.332399i\)
\(L(2)\) 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 \( 1 + (17.4 - 7.45i)T \)
good2 \( 1 + (-1.06 - 0.617i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (4.08 - 7.08i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 10.4T + 49T^{2} \)
11 \( 1 + 3.90T + 121T^{2} \)
13 \( 1 + (2.02 - 1.16i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-2.13 + 3.70i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-20.0 - 34.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-10.6 + 6.17i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 22.2iT - 961T^{2} \)
37 \( 1 + 62.2iT - 1.36e3T^{2} \)
41 \( 1 + (44.7 + 25.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-6.67 + 11.5i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-24.1 - 41.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (67.7 - 39.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-82.7 - 47.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-18.8 - 32.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (52.6 - 30.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (78.8 + 45.5i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-41.7 + 72.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-0.466 - 0.269i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 54.0T + 6.88e3T^{2} \)
89 \( 1 + (81.6 - 47.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (126. + 73.2i)T + (4.70e3 + 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.15631230131838618617760183659, −12.20584354479499628330154829330, −10.85365030114149522078060345420, −10.17964749643172023214056561987, −9.154368327513759690416169793507, −7.37640030953439258148113743009, −6.69990683763168600770073715575, −5.69627499676389327925964839249, −3.98120492427331931168961268045, −3.04390036225751004669657501590, 0.16441367624798546082912033082, 2.96145746912870371067596575831, 4.17583081577912541164362512876, 5.06473101769695880784597108832, 6.67770332209463277706484388378, 8.209457596196849383824364312084, 8.757004651285556158658704934929, 9.957664231443265542245069878036, 11.41607440739075013267021365007, 12.46325046143278250997532541224

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