Properties

Degree 2
Conductor $ 2^{3} \cdot 3 $
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 + 9-s + 4·11-s − 2·13-s + 2·15-s + 2·17-s − 4·19-s − 8·23-s − 25-s − 27-s + 6·29-s + 8·31-s − 4·33-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s − 2·45-s − 7·49-s − 2·51-s − 2·53-s − 8·55-s + 4·57-s + 4·59-s − 2·61-s + 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s − 49-s − 0.280·51-s − 0.274·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(24\)    =    \(2^{3} \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{24} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 24,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.539128$
$L(\frac12)$  $\approx$  $0.539128$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.67075195000178004295107787310, −16.62726100086658933765837426594, −15.47325164750050957705309114011, −14.18198826155077042068868714860, −12.31725686074733747032880351584, −11.58204746144152243175303945055, −9.964671915727173343349228861429, −8.098990694093691505068092710868, −6.42897107072744719896577758454, −4.25303028692796488061253338187, 4.25303028692796488061253338187, 6.42897107072744719896577758454, 8.098990694093691505068092710868, 9.964671915727173343349228861429, 11.58204746144152243175303945055, 12.31725686074733747032880351584, 14.18198826155077042068868714860, 15.47325164750050957705309114011, 16.62726100086658933765837426594, 17.67075195000178004295107787310

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