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  =  $(2,\ 96,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.00107$
$L(\frac12)$  $\approx$  $1.00107$
$L(\frac{3}{2})$   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

−13.87948688478378416848706681259, −13.02862772705358459630904610270, −11.67525385332011727244436354557, −10.84849442743057013529606410604, −9.826637660192491869357018654726, −8.391046533691483698004382219288, −7.17212008872630953090710242182, −5.56983356275681900785379754785, −4.77602747706575131165960838209, −2.09803110754033007053716721218, 2.09803110754033007053716721218, 4.77602747706575131165960838209, 5.56983356275681900785379754785, 7.17212008872630953090710242182, 8.391046533691483698004382219288, 9.826637660192491869357018654726, 10.84849442743057013529606410604, 11.67525385332011727244436354557, 13.02862772705358459630904610270, 13.87948688478378416848706681259

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