Properties

Label 2-1323-1.1-c3-0-113
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.78·2-s − 0.244·4-s − 20.8·5-s − 22.9·8-s − 58.0·10-s + 65.7·11-s + 50.9·13-s − 61.9·16-s + 46.1·17-s − 62.1·19-s + 5.09·20-s + 183.·22-s + 125.·23-s + 308.·25-s + 141.·26-s − 168.·29-s − 187.·31-s + 11.0·32-s + 128.·34-s − 70.0·37-s − 173.·38-s + 478.·40-s − 188.·41-s + 239.·43-s − 16.0·44-s + 349.·46-s − 591.·47-s + ⋯
L(s)  = 1  + 0.984·2-s − 0.0305·4-s − 1.86·5-s − 1.01·8-s − 1.83·10-s + 1.80·11-s + 1.08·13-s − 0.968·16-s + 0.658·17-s − 0.750·19-s + 0.0569·20-s + 1.77·22-s + 1.13·23-s + 2.47·25-s + 1.06·26-s − 1.08·29-s − 1.08·31-s + 0.0611·32-s + 0.647·34-s − 0.311·37-s − 0.738·38-s + 1.89·40-s − 0.717·41-s + 0.849·43-s − 0.0551·44-s + 1.11·46-s − 1.83·47-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: \(1\)
Selberg data: \((2,\ 1323,\ (\ :3/2),\ -1)\)

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)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - 2.78T + 8T^{2} \)
5 \( 1 + 20.8T + 125T^{2} \)
11 \( 1 - 65.7T + 1.33e3T^{2} \)
13 \( 1 - 50.9T + 2.19e3T^{2} \)
17 \( 1 - 46.1T + 4.91e3T^{2} \)
19 \( 1 + 62.1T + 6.85e3T^{2} \)
23 \( 1 - 125.T + 1.21e4T^{2} \)
29 \( 1 + 168.T + 2.43e4T^{2} \)
31 \( 1 + 187.T + 2.97e4T^{2} \)
37 \( 1 + 70.0T + 5.06e4T^{2} \)
41 \( 1 + 188.T + 6.89e4T^{2} \)
43 \( 1 - 239.T + 7.95e4T^{2} \)
47 \( 1 + 591.T + 1.03e5T^{2} \)
53 \( 1 + 464.T + 1.48e5T^{2} \)
59 \( 1 - 325.T + 2.05e5T^{2} \)
61 \( 1 - 261.T + 2.26e5T^{2} \)
67 \( 1 - 340.T + 3.00e5T^{2} \)
71 \( 1 + 752.T + 3.57e5T^{2} \)
73 \( 1 + 245.T + 3.89e5T^{2} \)
79 \( 1 - 546.T + 4.93e5T^{2} \)
83 \( 1 - 144.T + 5.71e5T^{2} \)
89 \( 1 - 603.T + 7.04e5T^{2} \)
97 \( 1 + 1.35e3T + 9.12e5T^{2} \)
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

−8.798452430197517993691375412984, −8.096267958396193294988654666962, −7.01371508083169232922537945529, −6.40052986392515194946825644506, −5.26385806026305456457525749333, −4.27739455200452039846331921678, −3.73041657947852908161071233173, −3.28962069901879926215964069627, −1.26073656780239051508675372815, 0, 1.26073656780239051508675372815, 3.28962069901879926215964069627, 3.73041657947852908161071233173, 4.27739455200452039846331921678, 5.26385806026305456457525749333, 6.40052986392515194946825644506, 7.01371508083169232922537945529, 8.096267958396193294988654666962, 8.798452430197517993691375412984

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