Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 7-s + 9-s − 4·11-s − 6·13-s − 2·15-s + 2·17-s − 4·19-s − 21-s − 8·23-s − 25-s − 27-s + 2·29-s + 4·33-s + 2·35-s + 10·37-s + 6·39-s − 6·41-s − 4·43-s + 2·45-s + 49-s − 2·51-s − 6·53-s − 8·55-s + 4·57-s + 4·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.338·35-s + 1.64·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-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: $\chi_{1344} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.90922528213177, −19.14984933025549, −18.21453356481622, −17.97880378114661, −17.12654746213102, −16.75878332738357, −15.86724947240857, −15.08236946943927, −14.40816684431301, −13.67871015109170, −12.90054066973290, −12.28662683288716, −11.56985295135314, −10.56233382470030, −10.02545149330825, −9.563538518206812, −8.186632228453794, −7.685015150188554, −6.608076600522026, −5.765369999194802, −5.117402880349692, −4.301676378496726, −2.667923367707886, −1.883513692797836, 0, 1.883513692797836, 2.667923367707886, 4.301676378496726, 5.117402880349692, 5.765369999194802, 6.608076600522026, 7.685015150188554, 8.186632228453794, 9.563538518206812, 10.02545149330825, 10.56233382470030, 11.56985295135314, 12.28662683288716, 12.90054066973290, 13.67871015109170, 14.40816684431301, 15.08236946943927, 15.86724947240857, 16.75878332738357, 17.12654746213102, 17.97880378114661, 18.21453356481622, 19.14984933025549, 19.90922528213177

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