Properties

Label 2-1344-1.1-c3-0-36
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 $1$

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: $-1$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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

−8.898388983103280563532982969003, −7.75314975862125950084012002168, −7.32790981849486330907346223526, −6.58572205966979815593786534839, −5.27507144677204813777710233267, −4.44228088863160655126473648652, −3.91109382706976883806622938279, −2.63765087168340060138996362987, −1.01237081630015100837115822755, 0, 1.01237081630015100837115822755, 2.63765087168340060138996362987, 3.91109382706976883806622938279, 4.44228088863160655126473648652, 5.27507144677204813777710233267, 6.58572205966979815593786534839, 7.32790981849486330907346223526, 7.75314975862125950084012002168, 8.898388983103280563532982969003

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