Properties

Degree $2$
Conductor $1344$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 18·5-s − 7·7-s + 9·9-s − 36·11-s + 34·13-s − 54·15-s + 42·17-s − 124·19-s + 21·21-s + 199·25-s − 27·27-s − 102·29-s + 160·31-s + 108·33-s − 126·35-s − 398·37-s − 102·39-s − 318·41-s − 268·43-s + 162·45-s − 240·47-s + 49·49-s − 126·51-s + 498·53-s − 648·55-s + 372·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.60·5-s − 0.377·7-s + 1/3·9-s − 0.986·11-s + 0.725·13-s − 0.929·15-s + 0.599·17-s − 1.49·19-s + 0.218·21-s + 1.59·25-s − 0.192·27-s − 0.653·29-s + 0.926·31-s + 0.569·33-s − 0.608·35-s − 1.76·37-s − 0.418·39-s − 1.21·41-s − 0.950·43-s + 0.536·45-s − 0.744·47-s + 1/7·49-s − 0.345·51-s + 1.29·53-s − 1.58·55-s + 0.864·57-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: $-1$
Motivic weight: \(3\)
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 \( 1 + p T \)
7 \( 1 + p T \)
good5 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 + 36 T + p^{3} T^{2} \)
13 \( 1 - 34 T + p^{3} T^{2} \)
17 \( 1 - 42 T + p^{3} T^{2} \)
19 \( 1 + 124 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + 102 T + p^{3} T^{2} \)
31 \( 1 - 160 T + p^{3} T^{2} \)
37 \( 1 + 398 T + p^{3} T^{2} \)
41 \( 1 + 318 T + p^{3} T^{2} \)
43 \( 1 + 268 T + p^{3} T^{2} \)
47 \( 1 + 240 T + p^{3} T^{2} \)
53 \( 1 - 498 T + p^{3} T^{2} \)
59 \( 1 + 132 T + p^{3} T^{2} \)
61 \( 1 + 398 T + p^{3} T^{2} \)
67 \( 1 - 92 T + p^{3} T^{2} \)
71 \( 1 - 720 T + p^{3} T^{2} \)
73 \( 1 + 502 T + p^{3} T^{2} \)
79 \( 1 - 1024 T + p^{3} T^{2} \)
83 \( 1 + 204 T + p^{3} T^{2} \)
89 \( 1 - 354 T + p^{3} T^{2} \)
97 \( 1 + 286 T + p^{3} T^{2} \)
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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

−8.908728494982113775020012382306, −8.157804292783529468695932224133, −6.84851962043389268036481184764, −6.29654902946747678865368645739, −5.54207436188813766556261803786, −4.91843027798274987127080857630, −3.52853598368426704556422443687, −2.34660792732589263082050553827, −1.46702406456527069752512268148, 0, 1.46702406456527069752512268148, 2.34660792732589263082050553827, 3.52853598368426704556422443687, 4.91843027798274987127080857630, 5.54207436188813766556261803786, 6.29654902946747678865368645739, 6.84851962043389268036481184764, 8.157804292783529468695932224133, 8.908728494982113775020012382306

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