Properties

Label 2-171-1.1-c1-0-5
Degree $2$
Conductor $171$
Sign $1$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 3·5-s − 5·7-s + 6·10-s − 11-s + 2·13-s − 10·14-s − 4·16-s + 17-s − 19-s + 6·20-s − 2·22-s + 4·23-s + 4·25-s + 4·26-s − 10·28-s + 2·29-s − 6·31-s − 8·32-s + 2·34-s − 15·35-s − 2·38-s − 43-s − 2·44-s + 8·46-s + 9·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.34·5-s − 1.88·7-s + 1.89·10-s − 0.301·11-s + 0.554·13-s − 2.67·14-s − 16-s + 0.242·17-s − 0.229·19-s + 1.34·20-s − 0.426·22-s + 0.834·23-s + 4/5·25-s + 0.784·26-s − 1.88·28-s + 0.371·29-s − 1.07·31-s − 1.41·32-s + 0.342·34-s − 2.53·35-s − 0.324·38-s − 0.152·43-s − 0.301·44-s + 1.17·46-s + 1.31·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.215096110\)
\(L(\frac12)\) \(\approx\) \(2.215096110\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−13.00980511664590576338005674082, −12.37438605696103795680934752253, −10.84385583714429238920836096770, −9.769325574936890193507770039971, −9.028755790922139382896168795846, −6.85073886350288365363080648917, −6.12241751609850681792173625996, −5.34668146252658900228619106104, −3.67840796781373175574076394185, −2.61687998381550726785939862479, 2.61687998381550726785939862479, 3.67840796781373175574076394185, 5.34668146252658900228619106104, 6.12241751609850681792173625996, 6.85073886350288365363080648917, 9.028755790922139382896168795846, 9.769325574936890193507770039971, 10.84385583714429238920836096770, 12.37438605696103795680934752253, 13.00980511664590576338005674082

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