Properties

Degree 2
Conductor $ 2^{5} \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 − 4·7-s + 9-s + 4·11-s − 2·13-s + 2·15-s − 6·17-s − 4·19-s − 4·21-s − 25-s + 27-s + 2·29-s + 4·31-s + 4·33-s − 8·35-s − 2·37-s − 2·39-s + 2·41-s + 4·43-s + 2·45-s + 8·47-s + 9·49-s − 6·51-s + 10·53-s + 8·55-s − 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 1.51·7-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 − 0.872·21-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.696·33-s − 1.35·35-s − 0.328·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.840·51-s + 1.37·53-s + 1.07·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(96\)    =    \(2^{5} \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{96} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 96,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.17142\)
\(L(\frac12)\)  \(\approx\)  \(1.17142\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.80073173066515229603635969273, −13.14140490363893974463899954609, −12.10541458487679021094810242454, −10.44913304188171401892835914004, −9.488699237661810053920246470193, −8.839886292016704738441339481672, −6.91469103626909406223255043804, −6.15267013015603292112906972931, −4.10625798014363516984369255368, −2.44821729992400328434125228202, 2.44821729992400328434125228202, 4.10625798014363516984369255368, 6.15267013015603292112906972931, 6.91469103626909406223255043804, 8.839886292016704738441339481672, 9.488699237661810053920246470193, 10.44913304188171401892835914004, 12.10541458487679021094810242454, 13.14140490363893974463899954609, 13.80073173066515229603635969273

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