Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s + 2·11-s + 2·13-s + 4·17-s + 4·19-s − 21-s − 6·23-s − 5·25-s + 27-s + 2·29-s + 2·33-s + 6·37-s + 2·39-s + 8·41-s + 8·43-s − 4·47-s + 49-s + 4·51-s + 6·53-s + 4·57-s + 14·61-s − 63-s − 4·67-s − 6·69-s − 2·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.970·17-s + 0.917·19-s − 0.218·21-s − 1.25·23-s − 25-s + 0.192·27-s + 0.371·29-s + 0.348·33-s + 0.986·37-s + 0.320·39-s + 1.24·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.560·51-s + 0.824·53-s + 0.529·57-s + 1.79·61-s − 0.125·63-s − 0.488·67-s − 0.722·69-s − 0.237·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1344} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.160376678\)
\(L(\frac12)\) \(\approx\) \(2.160376678\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + T \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−9.694432620383059295044846371230, −8.839846973587512271832482159777, −7.955699455902827737318339326897, −7.36235129594086681027946055340, −6.24035827483151455450786911728, −5.60729594063914370947972765374, −4.20599887184774343980959477769, −3.57055610765599388652202709940, −2.47132763276648395687043734661, −1.12641799956446265250447029021, 1.12641799956446265250447029021, 2.47132763276648395687043734661, 3.57055610765599388652202709940, 4.20599887184774343980959477769, 5.60729594063914370947972765374, 6.24035827483151455450786911728, 7.36235129594086681027946055340, 7.955699455902827737318339326897, 8.839846973587512271832482159777, 9.694432620383059295044846371230

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