Properties

Label 2-1344-1.1-c3-0-54
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 + 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: $-1$
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 \( 1 - 6.30T + 125T^{2} \)
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

−8.840960751031803822892154813561, −7.83948456397982464219365481451, −7.38221321660641608555739934424, −6.03831266585576194335398888208, −5.58935902445925575368396872861, −4.81755376541141603813152416760, −3.61237687761207064539602668607, −2.42319536327476058552345856761, −1.34352988145364680011807430776, 0, 1.34352988145364680011807430776, 2.42319536327476058552345856761, 3.61237687761207064539602668607, 4.81755376541141603813152416760, 5.58935902445925575368396872861, 6.03831266585576194335398888208, 7.38221321660641608555739934424, 7.83948456397982464219365481451, 8.840960751031803822892154813561

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