Properties

Label 2-2166-1.1-c3-0-155
Degree $2$
Conductor $2166$
Sign $-1$
Analytic cond. $127.798$
Root an. cond. $11.3047$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s + 4.18·5-s − 6·6-s + 3.18·7-s + 8·8-s + 9·9-s + 8.37·10-s + 69.4·11-s − 12·12-s − 8.12·13-s + 6.37·14-s − 12.5·15-s + 16·16-s − 106.·17-s + 18·18-s + 16.7·20-s − 9.56·21-s + 138.·22-s − 176.·23-s − 24·24-s − 107.·25-s − 16.2·26-s − 27·27-s + 12.7·28-s − 66.2·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.374·5-s − 0.408·6-s + 0.172·7-s + 0.353·8-s + 0.333·9-s + 0.264·10-s + 1.90·11-s − 0.288·12-s − 0.173·13-s + 0.121·14-s − 0.216·15-s + 0.250·16-s − 1.51·17-s + 0.235·18-s + 0.187·20-s − 0.0993·21-s + 1.34·22-s − 1.60·23-s − 0.204·24-s − 0.859·25-s − 0.122·26-s − 0.192·27-s + 0.0860·28-s − 0.424·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2166\)    =    \(2 \cdot 3 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(127.798\)
Root analytic conductor: \(11.3047\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2166} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2166,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + 3T \)
19 \( 1 \)
good5 \( 1 - 4.18T + 125T^{2} \)
7 \( 1 - 3.18T + 343T^{2} \)
11 \( 1 - 69.4T + 1.33e3T^{2} \)
13 \( 1 + 8.12T + 2.19e3T^{2} \)
17 \( 1 + 106.T + 4.91e3T^{2} \)
23 \( 1 + 176.T + 1.21e4T^{2} \)
29 \( 1 + 66.2T + 2.43e4T^{2} \)
31 \( 1 + 140.T + 2.97e4T^{2} \)
37 \( 1 + 156.T + 5.06e4T^{2} \)
41 \( 1 + 414.T + 6.89e4T^{2} \)
43 \( 1 - 115.T + 7.95e4T^{2} \)
47 \( 1 - 620.T + 1.03e5T^{2} \)
53 \( 1 + 371.T + 1.48e5T^{2} \)
59 \( 1 - 91.6T + 2.05e5T^{2} \)
61 \( 1 - 218.T + 2.26e5T^{2} \)
67 \( 1 + 145.T + 3.00e5T^{2} \)
71 \( 1 - 887.T + 3.57e5T^{2} \)
73 \( 1 - 199.T + 3.89e5T^{2} \)
79 \( 1 + 389.T + 4.93e5T^{2} \)
83 \( 1 - 380.T + 5.71e5T^{2} \)
89 \( 1 + 425.T + 7.04e5T^{2} \)
97 \( 1 - 419.T + 9.12e5T^{2} \)
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

−8.330200960335170579360266432947, −7.20783576111380342442012195739, −6.54928791323552567030951085607, −6.02844590069196671982796596644, −5.15981203909107988450380637301, −4.14985879806247405964625764716, −3.76419964941232294033542249397, −2.16132234553381211054735264234, −1.53016274877462214274435150437, 0, 1.53016274877462214274435150437, 2.16132234553381211054735264234, 3.76419964941232294033542249397, 4.14985879806247405964625764716, 5.15981203909107988450380637301, 6.02844590069196671982796596644, 6.54928791323552567030951085607, 7.20783576111380342442012195739, 8.330200960335170579360266432947

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