Properties

Label 2-1323-1.1-c1-0-46
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  + 2.64·2-s + 5.00·4-s + 2.64·5-s + 7.93·8-s + 7.00·10-s − 2.64·11-s + 2·13-s + 11.0·16-s − 7·19-s + 13.2·20-s − 7.00·22-s − 7.93·23-s + 2.00·25-s + 5.29·26-s + 5.29·29-s − 3·31-s + 13.2·32-s − 3·37-s − 18.5·38-s + 21.0·40-s − 2.64·41-s + 8·43-s − 13.2·44-s − 21.0·46-s + 5.29·50-s + 10.0·52-s − 7.00·55-s + ⋯
L(s)  = 1  + 1.87·2-s + 2.50·4-s + 1.18·5-s + 2.80·8-s + 2.21·10-s − 0.797·11-s + 0.554·13-s + 2.75·16-s − 1.60·19-s + 2.95·20-s − 1.49·22-s − 1.65·23-s + 0.400·25-s + 1.03·26-s + 0.982·29-s − 0.538·31-s + 2.33·32-s − 0.493·37-s − 3.00·38-s + 3.32·40-s − 0.413·41-s + 1.21·43-s − 1.99·44-s − 3.09·46-s + 0.748·50-s + 1.38·52-s − 0.943·55-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

\(L(1)\) \(\approx\) \(6.046265697\)
\(L(\frac12)\) \(\approx\) \(6.046265697\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64T + 2T^{2} \)
5 \( 1 - 2.64T + 5T^{2} \)
11 \( 1 + 2.64T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + 7.93T + 23T^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 + 2.64T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 7.93T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 - 18.5T + 89T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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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

−10.10298622004518825749087570464, −8.725418989418723089930402407623, −7.74443858311826301435970244497, −6.61972073495258015631834214141, −6.07355209842237419310298739082, −5.48769918369145578724784166699, −4.55752085290803469902278297907, −3.70742956508013399152369602234, −2.49286076782179912292461798588, −1.90129567862797964692328185767, 1.90129567862797964692328185767, 2.49286076782179912292461798588, 3.70742956508013399152369602234, 4.55752085290803469902278297907, 5.48769918369145578724784166699, 6.07355209842237419310298739082, 6.61972073495258015631834214141, 7.74443858311826301435970244497, 8.725418989418723089930402407623, 10.10298622004518825749087570464

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