Properties

Label 2-1323-1.1-c1-0-29
Degree $2$
Conductor $1323$
Sign $1$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.10·2-s + 2.41·4-s + 2.10·5-s + 0.870·8-s + 4.41·10-s − 3.84·11-s + 6.82·13-s − 2.99·16-s + 1.23·17-s + 4.41·19-s + 5.07·20-s − 8.07·22-s + 6.81·23-s − 0.585·25-s + 14.3·26-s + 8.40·29-s − 0.656·31-s − 8.04·32-s + 2.58·34-s + 3.24·37-s + 9.27·38-s + 1.82·40-s − 5.07·41-s − 7.07·43-s − 9.27·44-s + 14.3·46-s − 10.6·47-s + ⋯
L(s)  = 1  + 1.48·2-s + 1.20·4-s + 0.939·5-s + 0.307·8-s + 1.39·10-s − 1.15·11-s + 1.89·13-s − 0.749·16-s + 0.298·17-s + 1.01·19-s + 1.13·20-s − 1.72·22-s + 1.42·23-s − 0.117·25-s + 2.81·26-s + 1.56·29-s − 0.117·31-s − 1.42·32-s + 0.443·34-s + 0.533·37-s + 1.50·38-s + 0.289·40-s − 0.792·41-s − 1.07·43-s − 1.39·44-s + 2.11·46-s − 1.55·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.374101516\)
\(L(\frac12)\) \(\approx\) \(4.374101516\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.10T + 2T^{2} \)
5 \( 1 - 2.10T + 5T^{2} \)
11 \( 1 + 3.84T + 11T^{2} \)
13 \( 1 - 6.82T + 13T^{2} \)
17 \( 1 - 1.23T + 17T^{2} \)
19 \( 1 - 4.41T + 19T^{2} \)
23 \( 1 - 6.81T + 23T^{2} \)
29 \( 1 - 8.40T + 29T^{2} \)
31 \( 1 + 0.656T + 31T^{2} \)
37 \( 1 - 3.24T + 37T^{2} \)
41 \( 1 + 5.07T + 41T^{2} \)
43 \( 1 + 7.07T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 - 2.97T + 59T^{2} \)
61 \( 1 + 6.48T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 - 5.89T + 79T^{2} \)
83 \( 1 + 0.720T + 83T^{2} \)
89 \( 1 - 0.149T + 89T^{2} \)
97 \( 1 + 1.65T + 97T^{2} \)
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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

−9.751253030865841990199669321060, −8.814224060757540200405115880311, −7.931175386341359093646511434359, −6.68630451777558793754329236746, −6.11252333019577611237154748251, −5.32211948054130676557195257473, −4.75967814185732876595708699491, −3.41103462561610339595769797872, −2.87584121943402312033224766960, −1.46860582479803359462865164825, 1.46860582479803359462865164825, 2.87584121943402312033224766960, 3.41103462561610339595769797872, 4.75967814185732876595708699491, 5.32211948054130676557195257473, 6.11252333019577611237154748251, 6.68630451777558793754329236746, 7.931175386341359093646511434359, 8.814224060757540200405115880311, 9.751253030865841990199669321060

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