Properties

Label 2-1176-1.1-c3-0-1
Degree $2$
Conductor $1176$
Sign $1$
Analytic cond. $69.3862$
Root an. cond. $8.32984$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 11.4·5-s + 9·9-s − 52.5·11-s + 5.48·13-s + 34.3·15-s − 85.3·17-s + 110.·19-s − 209.·23-s + 5.99·25-s − 27·27-s − 132.·29-s + 49.3·31-s + 157.·33-s + 160.·37-s − 16.4·39-s − 138.·41-s − 365.·43-s − 103.·45-s − 131.·47-s + 256.·51-s − 561.·53-s + 601.·55-s − 331.·57-s + 436.·59-s + 291.·61-s − 62.7·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.02·5-s + 0.333·9-s − 1.44·11-s + 0.116·13-s + 0.591·15-s − 1.21·17-s + 1.33·19-s − 1.89·23-s + 0.0479·25-s − 0.192·27-s − 0.850·29-s + 0.285·31-s + 0.832·33-s + 0.712·37-s − 0.0675·39-s − 0.526·41-s − 1.29·43-s − 0.341·45-s − 0.407·47-s + 0.703·51-s − 1.45·53-s + 1.47·55-s − 0.771·57-s + 0.962·59-s + 0.612·61-s − 0.119·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(69.3862\)
Root analytic conductor: \(8.32984\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4399328906\)
\(L(\frac12)\) \(\approx\) \(0.4399328906\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 3T \)
7 \( 1 \)
good5 \( 1 + 11.4T + 125T^{2} \)
11 \( 1 + 52.5T + 1.33e3T^{2} \)
13 \( 1 - 5.48T + 2.19e3T^{2} \)
17 \( 1 + 85.3T + 4.91e3T^{2} \)
19 \( 1 - 110.T + 6.85e3T^{2} \)
23 \( 1 + 209.T + 1.21e4T^{2} \)
29 \( 1 + 132.T + 2.43e4T^{2} \)
31 \( 1 - 49.3T + 2.97e4T^{2} \)
37 \( 1 - 160.T + 5.06e4T^{2} \)
41 \( 1 + 138.T + 6.89e4T^{2} \)
43 \( 1 + 365.T + 7.95e4T^{2} \)
47 \( 1 + 131.T + 1.03e5T^{2} \)
53 \( 1 + 561.T + 1.48e5T^{2} \)
59 \( 1 - 436.T + 2.05e5T^{2} \)
61 \( 1 - 291.T + 2.26e5T^{2} \)
67 \( 1 + 593.T + 3.00e5T^{2} \)
71 \( 1 - 775.T + 3.57e5T^{2} \)
73 \( 1 + 330.T + 3.89e5T^{2} \)
79 \( 1 - 243.T + 4.93e5T^{2} \)
83 \( 1 + 332.T + 5.71e5T^{2} \)
89 \( 1 - 979.T + 7.04e5T^{2} \)
97 \( 1 + 466.T + 9.12e5T^{2} \)
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

−9.569637890404238493236897792017, −8.259155459801771210400630862181, −7.84864201007668684979293588796, −7.00174550936396784581466722126, −5.96876986041445407481915089934, −5.10341602566226007006706319083, −4.25659714257570027835040484075, −3.27829915966571566884209773419, −1.99193214617218706797531556679, −0.33504412875355610270142194740, 0.33504412875355610270142194740, 1.99193214617218706797531556679, 3.27829915966571566884209773419, 4.25659714257570027835040484075, 5.10341602566226007006706319083, 5.96876986041445407481915089934, 7.00174550936396784581466722126, 7.84864201007668684979293588796, 8.259155459801771210400630862181, 9.569637890404238493236897792017

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