Properties

Label 2-2166-1.1-c3-0-141
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 4·4-s + 7.24·5-s + 6·6-s + 30.2·7-s + 8·8-s + 9·9-s + 14.4·10-s + 47.4·11-s + 12·12-s + 70.6·13-s + 60.4·14-s + 21.7·15-s + 16·16-s − 29.4·17-s + 18·18-s + 28.9·20-s + 90.6·21-s + 94.8·22-s − 43.0·23-s + 24·24-s − 72.4·25-s + 141.·26-s + 27·27-s + 120.·28-s + 15.7·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.648·5-s + 0.408·6-s + 1.63·7-s + 0.353·8-s + 0.333·9-s + 0.458·10-s + 1.29·11-s + 0.288·12-s + 1.50·13-s + 1.15·14-s + 0.374·15-s + 0.250·16-s − 0.419·17-s + 0.235·18-s + 0.324·20-s + 0.941·21-s + 0.919·22-s − 0.389·23-s + 0.204·24-s − 0.579·25-s + 1.06·26-s + 0.192·27-s + 0.815·28-s + 0.101·29-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2166,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.853796767\)
\(L(\frac12)\) \(\approx\) \(7.853796767\)
\(L(\frac{5}{2})\) 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 \( 1 - 7.24T + 125T^{2} \)
7 \( 1 - 30.2T + 343T^{2} \)
11 \( 1 - 47.4T + 1.33e3T^{2} \)
13 \( 1 - 70.6T + 2.19e3T^{2} \)
17 \( 1 + 29.4T + 4.91e3T^{2} \)
23 \( 1 + 43.0T + 1.21e4T^{2} \)
29 \( 1 - 15.7T + 2.43e4T^{2} \)
31 \( 1 - 172.T + 2.97e4T^{2} \)
37 \( 1 + 291.T + 5.06e4T^{2} \)
41 \( 1 + 18.9T + 6.89e4T^{2} \)
43 \( 1 - 234.T + 7.95e4T^{2} \)
47 \( 1 - 264.T + 1.03e5T^{2} \)
53 \( 1 - 343.T + 1.48e5T^{2} \)
59 \( 1 + 718.T + 2.05e5T^{2} \)
61 \( 1 + 840.T + 2.26e5T^{2} \)
67 \( 1 + 898.T + 3.00e5T^{2} \)
71 \( 1 + 904.T + 3.57e5T^{2} \)
73 \( 1 + 212.T + 3.89e5T^{2} \)
79 \( 1 + 499.T + 4.93e5T^{2} \)
83 \( 1 + 784.T + 5.71e5T^{2} \)
89 \( 1 - 60.4T + 7.04e5T^{2} \)
97 \( 1 + 889.T + 9.12e5T^{2} \)
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

−8.702047439936899973286401318617, −8.010006851696793577658623203526, −7.11243464659254377459393289911, −6.18561703838415038060918090909, −5.62087088034140836336295156653, −4.40783338031470150506760061686, −4.08100574904459276470432531672, −2.88435458846259917307062202783, −1.63817157372709106295748638178, −1.41574074910932472181746502787, 1.41574074910932472181746502787, 1.63817157372709106295748638178, 2.88435458846259917307062202783, 4.08100574904459276470432531672, 4.40783338031470150506760061686, 5.62087088034140836336295156653, 6.18561703838415038060918090909, 7.11243464659254377459393289911, 8.010006851696793577658623203526, 8.702047439936899973286401318617

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