Properties

Degree $2$
Conductor $1344$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 6.30·5-s − 7·7-s + 9·9-s + 48.9·11-s + 2.60·13-s + 18.9·15-s + 136.·17-s + 45.2·19-s − 21·21-s + 38.1·23-s − 85.2·25-s + 27·27-s − 52.7·29-s + 14.7·31-s + 146.·33-s − 44.1·35-s − 333.·37-s + 7.82·39-s + 227.·41-s − 398.·43-s + 56.7·45-s + 184.·47-s + 49·49-s + 410.·51-s − 359.·53-s + 308.·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.563·5-s − 0.377·7-s + 0.333·9-s + 1.34·11-s + 0.0556·13-s + 0.325·15-s + 1.95·17-s + 0.545·19-s − 0.218·21-s + 0.345·23-s − 0.682·25-s + 0.192·27-s − 0.337·29-s + 0.0856·31-s + 0.774·33-s − 0.213·35-s − 1.48·37-s + 0.0321·39-s + 0.865·41-s − 1.41·43-s + 0.187·45-s + 0.572·47-s + 0.142·49-s + 1.12·51-s − 0.932·53-s + 0.755·55-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: $1$
Motivic weight: \(3\)
Character: $\chi_{1344} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.540846614\)
\(L(\frac12)\) \(\approx\) \(3.540846614\)
\(L(\frac{5}{2})\) 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 \( 1 - 6.30T + 125T^{2} \)
11 \( 1 - 48.9T + 1.33e3T^{2} \)
13 \( 1 - 2.60T + 2.19e3T^{2} \)
17 \( 1 - 136.T + 4.91e3T^{2} \)
19 \( 1 - 45.2T + 6.85e3T^{2} \)
23 \( 1 - 38.1T + 1.21e4T^{2} \)
29 \( 1 + 52.7T + 2.43e4T^{2} \)
31 \( 1 - 14.7T + 2.97e4T^{2} \)
37 \( 1 + 333.T + 5.06e4T^{2} \)
41 \( 1 - 227.T + 6.89e4T^{2} \)
43 \( 1 + 398.T + 7.95e4T^{2} \)
47 \( 1 - 184.T + 1.03e5T^{2} \)
53 \( 1 + 359.T + 1.48e5T^{2} \)
59 \( 1 - 99.9T + 2.05e5T^{2} \)
61 \( 1 - 674.T + 2.26e5T^{2} \)
67 \( 1 + 376.T + 3.00e5T^{2} \)
71 \( 1 - 1.18e3T + 3.57e5T^{2} \)
73 \( 1 + 735.T + 3.89e5T^{2} \)
79 \( 1 - 836.T + 4.93e5T^{2} \)
83 \( 1 - 293.T + 5.71e5T^{2} \)
89 \( 1 - 1.29e3T + 7.04e5T^{2} \)
97 \( 1 + 201.T + 9.12e5T^{2} \)
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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

−9.456843256988379242286409583716, −8.513283879156760018736000669286, −7.62864979551680821147675883056, −6.81770061740201026603753340311, −5.95474117321141810972155710062, −5.12344112340877457819572134439, −3.77271664871619328424593690309, −3.25600048542501633008371782151, −1.90649179891262216776572216233, −0.983320536246301504377058832985, 0.983320536246301504377058832985, 1.90649179891262216776572216233, 3.25600048542501633008371782151, 3.77271664871619328424593690309, 5.12344112340877457819572134439, 5.95474117321141810972155710062, 6.81770061740201026603753340311, 7.62864979551680821147675883056, 8.513283879156760018736000669286, 9.456843256988379242286409583716

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