Properties

Label 2-1323-21.20-c1-0-7
Degree $2$
Conductor $1323$
Sign $-0.409 + 0.912i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.75i·2-s − 5.61·4-s + 2.24·5-s − 9.97i·8-s + 6.19i·10-s + 0.797i·11-s + 5.06i·13-s + 16.2·16-s − 4.46·17-s − 1.25i·19-s − 12.6·20-s − 2.20·22-s + 3.78i·23-s + 0.0366·25-s − 13.9·26-s + ⋯
L(s)  = 1  + 1.95i·2-s − 2.80·4-s + 1.00·5-s − 3.52i·8-s + 1.95i·10-s + 0.240i·11-s + 1.40i·13-s + 4.07·16-s − 1.08·17-s − 0.287i·19-s − 2.81·20-s − 0.469·22-s + 0.789i·23-s + 0.00733·25-s − 2.73·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.409 + 0.912i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1322, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.409 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.013153786\)
\(L(\frac12)\) \(\approx\) \(1.013153786\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.75iT - 2T^{2} \)
5 \( 1 - 2.24T + 5T^{2} \)
11 \( 1 - 0.797iT - 11T^{2} \)
13 \( 1 - 5.06iT - 13T^{2} \)
17 \( 1 + 4.46T + 17T^{2} \)
19 \( 1 + 1.25iT - 19T^{2} \)
23 \( 1 - 3.78iT - 23T^{2} \)
29 \( 1 - 4.04iT - 29T^{2} \)
31 \( 1 - 5.35iT - 31T^{2} \)
37 \( 1 + 5.33T + 37T^{2} \)
41 \( 1 + 4.50T + 41T^{2} \)
43 \( 1 - 0.390T + 43T^{2} \)
47 \( 1 + 6.60T + 47T^{2} \)
53 \( 1 + 6.67iT - 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 - 8.12iT - 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 - 3.19iT - 71T^{2} \)
73 \( 1 + 9.03iT - 73T^{2} \)
79 \( 1 - 5.14T + 79T^{2} \)
83 \( 1 - 9.29T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + 1.93iT - 97T^{2} \)
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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

−9.645566969957592555037654643653, −9.146483561126256901803495368486, −8.589904317614626043531134742676, −7.50652795438602766802808743686, −6.66221094229418836482012378517, −6.41119593450548435333466732071, −5.25966867179630434578325266036, −4.75896578038184038155026595263, −3.65798952221618000978078440522, −1.73988765454096798368312236871, 0.41738980518762755904305415103, 1.80337820866212289761595151019, 2.57951280097128300443708685929, 3.50316475234936799584791368206, 4.57097589631195192053829371751, 5.41050450897024456382337795055, 6.24960269065203903985699299388, 7.950492450531561607633961448246, 8.601719507679066556542881523992, 9.512904631443609880878438125612

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