Properties

Label 2-210-1.1-c11-0-30
Degree $2$
Conductor $210$
Sign $1$
Analytic cond. $161.352$
Root an. cond. $12.7024$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 32·2-s + 243·3-s + 1.02e3·4-s + 3.12e3·5-s + 7.77e3·6-s + 1.68e4·7-s + 3.27e4·8-s + 5.90e4·9-s + 1.00e5·10-s + 6.98e5·11-s + 2.48e5·12-s + 2.29e5·13-s + 5.37e5·14-s + 7.59e5·15-s + 1.04e6·16-s + 9.77e6·17-s + 1.88e6·18-s + 1.20e7·19-s + 3.20e6·20-s + 4.08e6·21-s + 2.23e7·22-s − 5.57e7·23-s + 7.96e6·24-s + 9.76e6·25-s + 7.33e6·26-s + 1.43e7·27-s + 1.72e7·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.30·11-s + 0.288·12-s + 0.171·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s + 1.66·17-s + 0.235·18-s + 1.11·19-s + 0.223·20-s + 0.218·21-s + 0.924·22-s − 1.80·23-s + 0.204·24-s + 0.199·25-s + 0.121·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(161.352\)
Root analytic conductor: \(12.7024\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(6.819032083\)
\(L(\frac12)\) \(\approx\) \(6.819032083\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 32T \)
3 \( 1 - 243T \)
5 \( 1 - 3.12e3T \)
7 \( 1 - 1.68e4T \)
good11 \( 1 - 6.98e5T + 2.85e11T^{2} \)
13 \( 1 - 2.29e5T + 1.79e12T^{2} \)
17 \( 1 - 9.77e6T + 3.42e13T^{2} \)
19 \( 1 - 1.20e7T + 1.16e14T^{2} \)
23 \( 1 + 5.57e7T + 9.52e14T^{2} \)
29 \( 1 + 1.37e8T + 1.22e16T^{2} \)
31 \( 1 - 2.54e8T + 2.54e16T^{2} \)
37 \( 1 + 3.03e8T + 1.77e17T^{2} \)
41 \( 1 + 1.39e8T + 5.50e17T^{2} \)
43 \( 1 + 1.85e9T + 9.29e17T^{2} \)
47 \( 1 - 2.84e9T + 2.47e18T^{2} \)
53 \( 1 + 1.70e9T + 9.26e18T^{2} \)
59 \( 1 + 3.42e9T + 3.01e19T^{2} \)
61 \( 1 - 1.08e10T + 4.35e19T^{2} \)
67 \( 1 - 2.90e8T + 1.22e20T^{2} \)
71 \( 1 - 3.15e9T + 2.31e20T^{2} \)
73 \( 1 + 2.16e10T + 3.13e20T^{2} \)
79 \( 1 + 4.72e9T + 7.47e20T^{2} \)
83 \( 1 - 8.20e9T + 1.28e21T^{2} \)
89 \( 1 - 7.16e10T + 2.77e21T^{2} \)
97 \( 1 - 1.14e10T + 7.15e21T^{2} \)
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

−10.23761715670179917964592728929, −9.552068483188052170060468292267, −8.315998225678485834964672685107, −7.37859839777016282075531727108, −6.22044722763440522778596529484, −5.30568889524918928707048148403, −4.01101456240679535039804698886, −3.24520096290250700960999465739, −1.89770109376302234754972884116, −1.10630427223516749449499218581, 1.10630427223516749449499218581, 1.89770109376302234754972884116, 3.24520096290250700960999465739, 4.01101456240679535039804698886, 5.30568889524918928707048148403, 6.22044722763440522778596529484, 7.37859839777016282075531727108, 8.315998225678485834964672685107, 9.552068483188052170060468292267, 10.23761715670179917964592728929

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