Properties

Label 2-211-1.1-c1-0-4
Degree $2$
Conductor $211$
Sign $1$
Analytic cond. $1.68484$
Root an. cond. $1.29801$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s + 0.381·3-s − 1.61·4-s + 3.23·5-s − 0.236·6-s + 1.61·7-s + 2.23·8-s − 2.85·9-s − 2.00·10-s − 3·11-s − 0.618·12-s + 6.23·13-s − 1.00·14-s + 1.23·15-s + 1.85·16-s + 6.61·17-s + 1.76·18-s + 0.854·19-s − 5.23·20-s + 0.618·21-s + 1.85·22-s + 1.76·23-s + 0.854·24-s + 5.47·25-s − 3.85·26-s − 2.23·27-s − 2.61·28-s + ⋯
L(s)  = 1  − 0.437·2-s + 0.220·3-s − 0.809·4-s + 1.44·5-s − 0.0963·6-s + 0.611·7-s + 0.790·8-s − 0.951·9-s − 0.632·10-s − 0.904·11-s − 0.178·12-s + 1.72·13-s − 0.267·14-s + 0.319·15-s + 0.463·16-s + 1.60·17-s + 0.415·18-s + 0.195·19-s − 1.17·20-s + 0.134·21-s + 0.395·22-s + 0.367·23-s + 0.174·24-s + 1.09·25-s − 0.755·26-s − 0.430·27-s − 0.494·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(211\)
Sign: $1$
Analytic conductor: \(1.68484\)
Root analytic conductor: \(1.29801\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 211,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.127291067\)
\(L(\frac12)\) \(\approx\) \(1.127291067\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 - T \)
good2 \( 1 + 0.618T + 2T^{2} \)
3 \( 1 - 0.381T + 3T^{2} \)
5 \( 1 - 3.23T + 5T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 6.23T + 13T^{2} \)
17 \( 1 - 6.61T + 17T^{2} \)
19 \( 1 - 0.854T + 19T^{2} \)
23 \( 1 - 1.76T + 23T^{2} \)
29 \( 1 + 2.23T + 29T^{2} \)
31 \( 1 + 11.0T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 - 1.61T + 47T^{2} \)
53 \( 1 - 5.38T + 53T^{2} \)
59 \( 1 + 6.70T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 8.18T + 71T^{2} \)
73 \( 1 - 2.09T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 0.472T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 2.85T + 97T^{2} \)
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

−12.60765245875506820186987410092, −11.00946118260807241438039653299, −10.35441309180127155496159833846, −9.250352518181498519287741720120, −8.625227336884577484639808350636, −7.64720430681378228300193610192, −5.73413738679359382492653842031, −5.35409419253810527228981723930, −3.41443173535724879756555829508, −1.58038285544726488074611921013, 1.58038285544726488074611921013, 3.41443173535724879756555829508, 5.35409419253810527228981723930, 5.73413738679359382492653842031, 7.64720430681378228300193610192, 8.625227336884577484639808350636, 9.250352518181498519287741720120, 10.35441309180127155496159833846, 11.00946118260807241438039653299, 12.60765245875506820186987410092

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