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 + 2·5-s + 7-s + 9-s + 4·11-s + 2·13-s + 2·15-s − 6·17-s + 4·19-s + 21-s − 25-s + 27-s + 2·29-s + 4·33-s + 2·35-s − 6·37-s + 2·39-s + 2·41-s − 4·43-s + 2·45-s + 49-s − 6·51-s − 6·53-s + 8·55-s + 4·57-s + 12·59-s + 2·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s + 0.218·21-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.338·35-s − 0.986·37-s + 0.320·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s + 1.56·59-s + 0.256·61-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

\(L(1)\) \(\approx\) \(2.702547665\)
\(L(\frac12)\) \(\approx\) \(2.702547665\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−9.465508899880048812015346580650, −8.900585072204336118527974818427, −8.202224728141808387663165849679, −7.02967607454599654192884435004, −6.43796562380819911728572291466, −5.46214860425768039190452613646, −4.41591285682058366018514470855, −3.50387103718831454239019194187, −2.25227517080335123972207039631, −1.36358574500507065097493451168, 1.36358574500507065097493451168, 2.25227517080335123972207039631, 3.50387103718831454239019194187, 4.41591285682058366018514470855, 5.46214860425768039190452613646, 6.43796562380819911728572291466, 7.02967607454599654192884435004, 8.202224728141808387663165849679, 8.900585072204336118527974818427, 9.465508899880048812015346580650

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