↑

Properties

Label 1-3-3.2-r1-0-0
Degree $1$
Conductor $3$
Sign $1$
Analytic cond. $0.322394$
Root an. cond. $0.322394$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 11-s + 13-s − 14-s + 16-s − 17-s + 19-s − 20-s + 22-s − 23-s + 25-s − 26-s + 28-s − 29-s + 31-s − 32-s + 34-s − 35-s + 37-s − 38-s + 40-s − 41-s + ⋯
L(s)  = 1  − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 11-s + 13-s − 14-s + 16-s − 17-s + 19-s − 20-s + 22-s − 23-s + 25-s − 26-s + 28-s − 29-s + 31-s − 32-s + 34-s − 35-s + 37-s − 38-s + 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3\)
Sign: $1$
Analytic conductor: \(0.322394\)
Root analytic conductor: \(0.322394\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 3,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4808675576\)
\(L(\frac12)\) \(\approx\) \(0.4808675576\)
\(L(1)\) \(\approx\) \(0.6045997880\)
\(L(1)\) \(\approx\) \(0.6045997880\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−63.17699276733910829118311838811, −62.20607812228213648686964588013, −60.02685745671590223526149077217, −57.58405636044918378148658948822, −55.6425587002162213738564033666, −54.19384310155191357860186830353, −52.4967495990607536672882221717, −50.375138650636265982821917341449, −47.5141045101173221149363569968, −46.27411802351314008510487166014, −44.12057291207220244202440459309, −42.6163792261575675743228068514, −39.48520726092935076736629266003, −37.55179655636462701986044645770, −35.60841265393863465481291012803, −33.89738892725941901767787403300, −30.74504026138249573780824181050, −28.21816450623338609318302976031, −26.57786873577458531458435093753, −24.05941485649345077459305359321, −20.45577080774249285344502583131, −18.26199749569312756892441409359, −15.70461917672162556516555088043, −11.24920620777293524970502567886, −8.03973715568146668171362321417, 8.03973715568146668171362321417, 11.24920620777293524970502567886, 15.70461917672162556516555088043, 18.26199749569312756892441409359, 20.45577080774249285344502583131, 24.05941485649345077459305359321, 26.57786873577458531458435093753, 28.21816450623338609318302976031, 30.74504026138249573780824181050, 33.89738892725941901767787403300, 35.60841265393863465481291012803, 37.55179655636462701986044645770, 39.48520726092935076736629266003, 42.6163792261575675743228068514, 44.12057291207220244202440459309, 46.27411802351314008510487166014, 47.5141045101173221149363569968, 50.375138650636265982821917341449, 52.4967495990607536672882221717, 54.19384310155191357860186830353, 55.6425587002162213738564033666, 57.58405636044918378148658948822, 60.02685745671590223526149077217, 62.20607812228213648686964588013, 63.17699276733910829118311838811

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

The lowest zero of this L-function, at height approximately 8.039, is the second highest among degree 1 L-functions.