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 − 20.3·5-s − 7·7-s + 9·9-s − 30.9·11-s − 50.6·13-s − 60.9·15-s − 102.·17-s − 61.2·19-s − 21·21-s − 148.·23-s + 287.·25-s + 27·27-s − 159.·29-s + 121.·31-s − 92.7·33-s + 142.·35-s + 357.·37-s − 151.·39-s + 466.·41-s − 185.·43-s − 182.·45-s + 131.·47-s + 49·49-s − 308.·51-s − 200.·53-s + 627.·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.81·5-s − 0.377·7-s + 0.333·9-s − 0.847·11-s − 1.07·13-s − 1.04·15-s − 1.46·17-s − 0.739·19-s − 0.218·21-s − 1.34·23-s + 2.29·25-s + 0.192·27-s − 1.01·29-s + 0.702·31-s − 0.489·33-s + 0.686·35-s + 1.59·37-s − 0.623·39-s + 1.77·41-s − 0.658·43-s − 0.605·45-s + 0.407·47-s + 0.142·49-s − 0.846·51-s − 0.518·53-s + 1.53·55-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

\(L(2)\) \(\approx\) \(0.4466545577\)
\(L(\frac12)\) \(\approx\) \(0.4466545577\)
\(L(\frac{5}{2})\) 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 \( 1 + 20.3T + 125T^{2} \)
11 \( 1 + 30.9T + 1.33e3T^{2} \)
13 \( 1 + 50.6T + 2.19e3T^{2} \)
17 \( 1 + 102.T + 4.91e3T^{2} \)
19 \( 1 + 61.2T + 6.85e3T^{2} \)
23 \( 1 + 148.T + 1.21e4T^{2} \)
29 \( 1 + 159.T + 2.43e4T^{2} \)
31 \( 1 - 121.T + 2.97e4T^{2} \)
37 \( 1 - 357.T + 5.06e4T^{2} \)
41 \( 1 - 466.T + 6.89e4T^{2} \)
43 \( 1 + 185.T + 7.95e4T^{2} \)
47 \( 1 - 131.T + 1.03e5T^{2} \)
53 \( 1 + 200.T + 1.48e5T^{2} \)
59 \( 1 + 591.T + 2.05e5T^{2} \)
61 \( 1 + 70.5T + 2.26e5T^{2} \)
67 \( 1 + 643.T + 3.00e5T^{2} \)
71 \( 1 - 522.T + 3.57e5T^{2} \)
73 \( 1 + 576.T + 3.89e5T^{2} \)
79 \( 1 + 280.T + 4.93e5T^{2} \)
83 \( 1 + 557.T + 5.71e5T^{2} \)
89 \( 1 + 1.22e3T + 7.04e5T^{2} \)
97 \( 1 - 65.0T + 9.12e5T^{2} \)
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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

−9.103484428239763757380815784192, −8.255310701113584102055737365140, −7.70210732346850764069569983535, −7.12813488032768843761859641876, −6.04101089281835508892110668458, −4.44924565826895101859961790184, −4.32126864381963202747234487950, −3.07070072674317224582188115469, −2.26689847740700276137625739031, −0.30179698712265435674489547231, 0.30179698712265435674489547231, 2.26689847740700276137625739031, 3.07070072674317224582188115469, 4.32126864381963202747234487950, 4.44924565826895101859961790184, 6.04101089281835508892110668458, 7.12813488032768843761859641876, 7.70210732346850764069569983535, 8.255310701113584102055737365140, 9.103484428239763757380815784192

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