Properties

Label 2-1323-1.1-c3-0-23
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.55·2-s − 5.59·4-s − 9.81·5-s − 21.0·8-s − 15.2·10-s + 70.6·11-s − 55.5·13-s + 12.0·16-s − 13.4·17-s − 91.3·19-s + 54.8·20-s + 109.·22-s − 113.·23-s − 28.7·25-s − 86.1·26-s − 12.4·29-s − 222.·31-s + 187.·32-s − 20.8·34-s + 257.·37-s − 141.·38-s + 206.·40-s − 286.·41-s + 4.81·43-s − 394.·44-s − 176.·46-s + 609.·47-s + ⋯
L(s)  = 1  + 0.548·2-s − 0.699·4-s − 0.877·5-s − 0.931·8-s − 0.481·10-s + 1.93·11-s − 1.18·13-s + 0.188·16-s − 0.191·17-s − 1.10·19-s + 0.613·20-s + 1.06·22-s − 1.03·23-s − 0.229·25-s − 0.650·26-s − 0.0800·29-s − 1.29·31-s + 1.03·32-s − 0.104·34-s + 1.14·37-s − 0.605·38-s + 0.817·40-s − 1.09·41-s + 0.0170·43-s − 1.35·44-s − 0.565·46-s + 1.89·47-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: $\chi_{1323} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.182476888\)
\(L(\frac12)\) \(\approx\) \(1.182476888\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.55T + 8T^{2} \)
5 \( 1 + 9.81T + 125T^{2} \)
11 \( 1 - 70.6T + 1.33e3T^{2} \)
13 \( 1 + 55.5T + 2.19e3T^{2} \)
17 \( 1 + 13.4T + 4.91e3T^{2} \)
19 \( 1 + 91.3T + 6.85e3T^{2} \)
23 \( 1 + 113.T + 1.21e4T^{2} \)
29 \( 1 + 12.4T + 2.43e4T^{2} \)
31 \( 1 + 222.T + 2.97e4T^{2} \)
37 \( 1 - 257.T + 5.06e4T^{2} \)
41 \( 1 + 286.T + 6.89e4T^{2} \)
43 \( 1 - 4.81T + 7.95e4T^{2} \)
47 \( 1 - 609.T + 1.03e5T^{2} \)
53 \( 1 + 691.T + 1.48e5T^{2} \)
59 \( 1 + 217.T + 2.05e5T^{2} \)
61 \( 1 - 764.T + 2.26e5T^{2} \)
67 \( 1 + 98.5T + 3.00e5T^{2} \)
71 \( 1 - 921.T + 3.57e5T^{2} \)
73 \( 1 - 219.T + 3.89e5T^{2} \)
79 \( 1 + 9.42T + 4.93e5T^{2} \)
83 \( 1 - 800.T + 5.71e5T^{2} \)
89 \( 1 - 253.T + 7.04e5T^{2} \)
97 \( 1 - 59.2T + 9.12e5T^{2} \)
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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

−9.268491110835052559365642046344, −8.504644085997278823099769013714, −7.65539286313107260325773739977, −6.69081232173316504816994556883, −5.91351902338233832174905264054, −4.74994257986462456538914680011, −4.07000984457363537420500899047, −3.57523787172336859884453511744, −2.06755728409460973295178542563, −0.49306999778494320234740779790, 0.49306999778494320234740779790, 2.06755728409460973295178542563, 3.57523787172336859884453511744, 4.07000984457363537420500899047, 4.74994257986462456538914680011, 5.91351902338233832174905264054, 6.69081232173316504816994556883, 7.65539286313107260325773739977, 8.504644085997278823099769013714, 9.268491110835052559365642046344

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