Properties

Label 2-1323-1.1-c3-0-108
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.77·2-s + 14.7·4-s + 1.85·5-s − 32.3·8-s − 8.87·10-s + 13.8·11-s + 74.4·13-s + 36.1·16-s − 89.3·17-s + 50.0·19-s + 27.4·20-s − 66.0·22-s − 33.2·23-s − 121.·25-s − 355.·26-s − 18.7·29-s + 52.1·31-s + 86.2·32-s + 426.·34-s − 152.·37-s − 238.·38-s − 60.1·40-s + 240.·41-s + 297.·43-s + 204.·44-s + 158.·46-s − 520.·47-s + ⋯
L(s)  = 1  − 1.68·2-s + 1.84·4-s + 0.166·5-s − 1.42·8-s − 0.280·10-s + 0.379·11-s + 1.58·13-s + 0.564·16-s − 1.27·17-s + 0.604·19-s + 0.307·20-s − 0.640·22-s − 0.301·23-s − 0.972·25-s − 2.67·26-s − 0.119·29-s + 0.302·31-s + 0.476·32-s + 2.15·34-s − 0.676·37-s − 1.01·38-s − 0.237·40-s + 0.914·41-s + 1.05·43-s + 0.700·44-s + 0.509·46-s − 1.61·47-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 \( 1 + 4.77T + 8T^{2} \)
5 \( 1 - 1.85T + 125T^{2} \)
11 \( 1 - 13.8T + 1.33e3T^{2} \)
13 \( 1 - 74.4T + 2.19e3T^{2} \)
17 \( 1 + 89.3T + 4.91e3T^{2} \)
19 \( 1 - 50.0T + 6.85e3T^{2} \)
23 \( 1 + 33.2T + 1.21e4T^{2} \)
29 \( 1 + 18.7T + 2.43e4T^{2} \)
31 \( 1 - 52.1T + 2.97e4T^{2} \)
37 \( 1 + 152.T + 5.06e4T^{2} \)
41 \( 1 - 240.T + 6.89e4T^{2} \)
43 \( 1 - 297.T + 7.95e4T^{2} \)
47 \( 1 + 520.T + 1.03e5T^{2} \)
53 \( 1 + 566.T + 1.48e5T^{2} \)
59 \( 1 + 622.T + 2.05e5T^{2} \)
61 \( 1 + 714.T + 2.26e5T^{2} \)
67 \( 1 + 45.3T + 3.00e5T^{2} \)
71 \( 1 + 865.T + 3.57e5T^{2} \)
73 \( 1 - 966.T + 3.89e5T^{2} \)
79 \( 1 - 1.01e3T + 4.93e5T^{2} \)
83 \( 1 + 380.T + 5.71e5T^{2} \)
89 \( 1 + 104.T + 7.04e5T^{2} \)
97 \( 1 - 765.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.025751236145680913612974146989, −8.178014408605403431819724597758, −7.54320776742565532187510046518, −6.47095791381693706461218772838, −6.05289579897806543288219268698, −4.51163362876657725221749890614, −3.32929635904348836602273141533, −2.01642472667265861816244478019, −1.21962719848813228380854939322, 0, 1.21962719848813228380854939322, 2.01642472667265861816244478019, 3.32929635904348836602273141533, 4.51163362876657725221749890614, 6.05289579897806543288219268698, 6.47095791381693706461218772838, 7.54320776742565532187510046518, 8.178014408605403431819724597758, 9.025751236145680913612974146989

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