Properties

Label 2-1323-1.1-c3-0-62
Degree $2$
Conductor $1323$
Sign $1$
Analytic cond. $78.0595$
Root an. cond. $8.83513$
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  + 0.560·2-s − 7.68·4-s + 12.9·5-s − 8.79·8-s + 7.26·10-s + 48.2·11-s + 36.3·13-s + 56.5·16-s + 83.2·17-s − 67.4·19-s − 99.5·20-s + 27.0·22-s − 30.6·23-s + 42.8·25-s + 20.3·26-s + 294.·29-s − 270.·31-s + 102.·32-s + 46.7·34-s − 204.·37-s − 37.8·38-s − 114.·40-s + 287.·41-s − 55.4·43-s − 370.·44-s − 17.1·46-s − 191.·47-s + ⋯
L(s)  = 1  + 0.198·2-s − 0.960·4-s + 1.15·5-s − 0.388·8-s + 0.229·10-s + 1.32·11-s + 0.774·13-s + 0.883·16-s + 1.18·17-s − 0.814·19-s − 1.11·20-s + 0.262·22-s − 0.277·23-s + 0.342·25-s + 0.153·26-s + 1.88·29-s − 1.56·31-s + 0.564·32-s + 0.235·34-s − 0.907·37-s − 0.161·38-s − 0.450·40-s + 1.09·41-s − 0.196·43-s − 1.26·44-s − 0.0550·46-s − 0.593·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(78.0595\)
Root analytic conductor: \(8.83513\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.762725439\)
\(L(\frac12)\) \(\approx\) \(2.762725439\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - 0.560T + 8T^{2} \)
5 \( 1 - 12.9T + 125T^{2} \)
11 \( 1 - 48.2T + 1.33e3T^{2} \)
13 \( 1 - 36.3T + 2.19e3T^{2} \)
17 \( 1 - 83.2T + 4.91e3T^{2} \)
19 \( 1 + 67.4T + 6.85e3T^{2} \)
23 \( 1 + 30.6T + 1.21e4T^{2} \)
29 \( 1 - 294.T + 2.43e4T^{2} \)
31 \( 1 + 270.T + 2.97e4T^{2} \)
37 \( 1 + 204.T + 5.06e4T^{2} \)
41 \( 1 - 287.T + 6.89e4T^{2} \)
43 \( 1 + 55.4T + 7.95e4T^{2} \)
47 \( 1 + 191.T + 1.03e5T^{2} \)
53 \( 1 - 521.T + 1.48e5T^{2} \)
59 \( 1 + 381.T + 2.05e5T^{2} \)
61 \( 1 + 155.T + 2.26e5T^{2} \)
67 \( 1 + 65.1T + 3.00e5T^{2} \)
71 \( 1 + 256.T + 3.57e5T^{2} \)
73 \( 1 + 318.T + 3.89e5T^{2} \)
79 \( 1 - 77.7T + 4.93e5T^{2} \)
83 \( 1 - 836.T + 5.71e5T^{2} \)
89 \( 1 - 1.59e3T + 7.04e5T^{2} \)
97 \( 1 - 1.18e3T + 9.12e5T^{2} \)
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

−9.062133135395288912530015842991, −8.850739853497396173990246645391, −7.73454635949751500005943355278, −6.43634640121443794978240275124, −5.97526350594839504105123209844, −5.09723553435956171734116182726, −4.08822225425534827987252077575, −3.29939362854787232810385678014, −1.81424326813261366354153769407, −0.873694399173240745216618664997, 0.873694399173240745216618664997, 1.81424326813261366354153769407, 3.29939362854787232810385678014, 4.08822225425534827987252077575, 5.09723553435956171734116182726, 5.97526350594839504105123209844, 6.43634640121443794978240275124, 7.73454635949751500005943355278, 8.850739853497396173990246645391, 9.062133135395288912530015842991

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