Properties

Label 2-1344-1.1-c3-0-1
Degree $2$
Conductor $1344$
Sign $1$
Analytic cond. $79.2985$
Root an. cond. $8.90497$
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  − 3·3-s − 15.3·5-s − 7·7-s + 9·9-s − 17.9·11-s + 17.3·13-s + 46.1·15-s − 84.3·17-s − 159.·19-s + 21·21-s + 115.·23-s + 111.·25-s − 27·27-s − 215.·29-s − 144.·31-s + 53.9·33-s + 107.·35-s − 150.·37-s − 52.1·39-s − 229.·41-s − 411.·43-s − 138.·45-s + 45.2·47-s + 49·49-s + 253.·51-s + 604.·53-s + 277.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.37·5-s − 0.377·7-s + 0.333·9-s − 0.493·11-s + 0.371·13-s + 0.794·15-s − 1.20·17-s − 1.92·19-s + 0.218·21-s + 1.05·23-s + 0.895·25-s − 0.192·27-s − 1.37·29-s − 0.835·31-s + 0.284·33-s + 0.520·35-s − 0.670·37-s − 0.214·39-s − 0.874·41-s − 1.45·43-s − 0.458·45-s + 0.140·47-s + 0.142·49-s + 0.694·51-s + 1.56·53-s + 0.679·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $1$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1587279410\)
\(L(\frac12)\) \(\approx\) \(0.1587279410\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 3T \)
7 \( 1 + 7T \)
good5 \( 1 + 15.3T + 125T^{2} \)
11 \( 1 + 17.9T + 1.33e3T^{2} \)
13 \( 1 - 17.3T + 2.19e3T^{2} \)
17 \( 1 + 84.3T + 4.91e3T^{2} \)
19 \( 1 + 159.T + 6.85e3T^{2} \)
23 \( 1 - 115.T + 1.21e4T^{2} \)
29 \( 1 + 215.T + 2.43e4T^{2} \)
31 \( 1 + 144.T + 2.97e4T^{2} \)
37 \( 1 + 150.T + 5.06e4T^{2} \)
41 \( 1 + 229.T + 6.89e4T^{2} \)
43 \( 1 + 411.T + 7.95e4T^{2} \)
47 \( 1 - 45.2T + 1.03e5T^{2} \)
53 \( 1 - 604.T + 1.48e5T^{2} \)
59 \( 1 + 315.T + 2.05e5T^{2} \)
61 \( 1 + 595.T + 2.26e5T^{2} \)
67 \( 1 + 311.T + 3.00e5T^{2} \)
71 \( 1 + 358.T + 3.57e5T^{2} \)
73 \( 1 + 816.T + 3.89e5T^{2} \)
79 \( 1 + 137.T + 4.93e5T^{2} \)
83 \( 1 + 64.5T + 5.71e5T^{2} \)
89 \( 1 - 487.T + 7.04e5T^{2} \)
97 \( 1 - 1.28e3T + 9.12e5T^{2} \)
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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

−8.964963316373908878996355485549, −8.552633169797653336044227226864, −7.46274820335089144184036539590, −6.88492762520492514702679321174, −6.00865899900285077822326902680, −4.87664844951692625282179973232, −4.14243795160563601444384945171, −3.30808484270123979424960430841, −1.90744709622484546793163572686, −0.19566998075658332241712031389, 0.19566998075658332241712031389, 1.90744709622484546793163572686, 3.30808484270123979424960430841, 4.14243795160563601444384945171, 4.87664844951692625282179973232, 6.00865899900285077822326902680, 6.88492762520492514702679321174, 7.46274820335089144184036539590, 8.552633169797653336044227226864, 8.964963316373908878996355485549

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