↑

Properties

Label 2-171-171.106-c1-0-1
Degree $2$
Conductor $171$
Sign $0.570 - 0.821i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.978 − 1.69i)2-s + (−1.73 − 0.0795i)3-s + (−0.914 + 1.58i)4-s − 0.196·5-s + (1.55 + 3.01i)6-s + (−2.23 + 3.86i)7-s − 0.332·8-s + (2.98 + 0.275i)9-s + (0.192 + 0.332i)10-s + (0.0755 − 0.130i)11-s + (1.70 − 2.66i)12-s + (−0.234 + 0.406i)13-s + 8.74·14-s + (0.339 + 0.0156i)15-s + (2.15 + 3.73i)16-s + (0.441 − 0.764i)17-s + ⋯
L(s)  = 1  + (−0.691 − 1.19i)2-s + (−0.998 − 0.0459i)3-s + (−0.457 + 0.792i)4-s − 0.0877·5-s + (0.636 + 1.22i)6-s + (−0.844 + 1.46i)7-s − 0.117·8-s + (0.995 + 0.0917i)9-s + (0.0607 + 0.105i)10-s + (0.0227 − 0.0394i)11-s + (0.493 − 0.770i)12-s + (−0.0650 + 0.112i)13-s + 2.33·14-s + (0.0876 + 0.00402i)15-s + (0.538 + 0.933i)16-s + (0.107 − 0.185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.570 - 0.821i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.570 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233356 + 0.121978i\)
\(L(\frac12)\) \(\approx\) \(0.233356 + 0.121978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0795i)T \)
19 \( 1 + (0.132 - 4.35i)T \)
good2 \( 1 + (0.978 + 1.69i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 0.196T + 5T^{2} \)
7 \( 1 + (2.23 - 3.86i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0755 + 0.130i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.234 - 0.406i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.441 + 0.764i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.57 - 4.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.57T + 29T^{2} \)
31 \( 1 + (1.37 + 2.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.36T + 37T^{2} \)
41 \( 1 + 8.13T + 41T^{2} \)
43 \( 1 + (-4.09 - 7.09i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 + (-5.86 - 10.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.58T + 59T^{2} \)
61 \( 1 - 9.13T + 61T^{2} \)
67 \( 1 + (-6.46 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.90 + 6.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.72 + 9.91i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.964 - 1.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.40 - 2.43i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.84 - 8.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.61 + 13.1i)T + (-48.5 + 84.0i)T^{2} \)
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

−12.27329123440341668637657752975, −11.94872779044368633091292553345, −11.06093486742258202632230059836, −9.840669439735229470148635682628, −9.452686228215296722026095073680, −8.057092009818601342807886100398, −6.32750715699758741710061431041, −5.52799356875309405098715489928, −3.55485861493082110414760378203, −1.94715801462528487278788247415, 0.33370232815107995592194707497, 3.88921274937118640684405196172, 5.38652416033291191569483623420, 6.72169365810968568711089857506, 6.98737391749677990968859322494, 8.246944560834746443359941630969, 9.710221650251483133707747248534, 10.29291439625311549035189353007, 11.48091325196090926499849942118, 12.68085236678829182172227424675

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