Properties

Label 2-1344-1.1-c3-0-38
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 + 2.63·5-s + 7·7-s + 9·9-s + 24.4·11-s + 83.0·13-s + 7.89·15-s + 110.·17-s − 26.7·19-s + 21·21-s + 50.1·23-s − 118.·25-s + 27·27-s + 16.7·29-s + 129.·31-s + 73.2·33-s + 18.4·35-s − 122.·37-s + 249.·39-s − 417.·41-s − 153.·43-s + 23.6·45-s − 175.·47-s + 49·49-s + 330.·51-s + 487.·53-s + 64.3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.235·5-s + 0.377·7-s + 0.333·9-s + 0.669·11-s + 1.77·13-s + 0.135·15-s + 1.57·17-s − 0.322·19-s + 0.218·21-s + 0.454·23-s − 0.944·25-s + 0.192·27-s + 0.107·29-s + 0.748·31-s + 0.386·33-s + 0.0890·35-s − 0.543·37-s + 1.02·39-s − 1.59·41-s − 0.543·43-s + 0.0785·45-s − 0.545·47-s + 0.142·49-s + 0.907·51-s + 1.26·53-s + 0.157·55-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

\(L(2)\) \(\approx\) \(3.746537649\)
\(L(\frac12)\) \(\approx\) \(3.746537649\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.63T + 125T^{2} \)
11 \( 1 - 24.4T + 1.33e3T^{2} \)
13 \( 1 - 83.0T + 2.19e3T^{2} \)
17 \( 1 - 110.T + 4.91e3T^{2} \)
19 \( 1 + 26.7T + 6.85e3T^{2} \)
23 \( 1 - 50.1T + 1.21e4T^{2} \)
29 \( 1 - 16.7T + 2.43e4T^{2} \)
31 \( 1 - 129.T + 2.97e4T^{2} \)
37 \( 1 + 122.T + 5.06e4T^{2} \)
41 \( 1 + 417.T + 6.89e4T^{2} \)
43 \( 1 + 153.T + 7.95e4T^{2} \)
47 \( 1 + 175.T + 1.03e5T^{2} \)
53 \( 1 - 487.T + 1.48e5T^{2} \)
59 \( 1 + 496.T + 2.05e5T^{2} \)
61 \( 1 + 274.T + 2.26e5T^{2} \)
67 \( 1 + 396.T + 3.00e5T^{2} \)
71 \( 1 - 844.T + 3.57e5T^{2} \)
73 \( 1 - 895.T + 3.89e5T^{2} \)
79 \( 1 - 1.31e3T + 4.93e5T^{2} \)
83 \( 1 + 1.36e3T + 5.71e5T^{2} \)
89 \( 1 - 683.T + 7.04e5T^{2} \)
97 \( 1 - 1.43e3T + 9.12e5T^{2} \)
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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

−9.141006191732136374393165937669, −8.407800751178726903629288629272, −7.86018976992273219387225974429, −6.73056902149020834284662397160, −6.00821644804223754090133028911, −5.03324126463399057870108541397, −3.82984513460898188227861727302, −3.28185249221774392209413719423, −1.82339781465875641341908057288, −1.04470294321620829505319448805, 1.04470294321620829505319448805, 1.82339781465875641341908057288, 3.28185249221774392209413719423, 3.82984513460898188227861727302, 5.03324126463399057870108541397, 6.00821644804223754090133028911, 6.73056902149020834284662397160, 7.86018976992273219387225974429, 8.407800751178726903629288629272, 9.141006191732136374393165937669

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