Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5 $
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 − 5-s + 9-s + 4·11-s + 2·13-s − 15-s + 2·17-s − 4·19-s + 25-s + 27-s + 2·29-s + 4·33-s + 10·37-s + 2·39-s + 10·41-s − 4·43-s − 45-s + 8·47-s − 7·49-s + 2·51-s + 10·53-s − 4·55-s − 4·57-s + 4·59-s + 2·61-s − 2·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 1.64·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.280·51-s + 1.37·53-s − 0.539·55-s − 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.248·65-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{960} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 960,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.980751817$
$L(\frac12)$  $\approx$  $1.980751817$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
good7 \( 1 + p T^{2} \)
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 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−19.61684514044677, −19.05224813678882, −18.29741129905900, −17.49918157977795, −16.62868764927288, −16.14844729407519, −15.12059497497638, −14.70088179223446, −13.98076737731680, −13.13885481122960, −12.39561579952124, −11.60902251767536, −10.87392441792526, −9.907104671360378, −9.080341213166513, −8.438902886313496, −7.601251803566526, −6.678759014069607, −5.851306659918507, −4.388717395027862, −3.823491730752669, −2.632525401750877, −1.188153900302878, 1.188153900302878, 2.632525401750877, 3.823491730752669, 4.388717395027862, 5.851306659918507, 6.678759014069607, 7.601251803566526, 8.438902886313496, 9.080341213166513, 9.907104671360378, 10.87392441792526, 11.60902251767536, 12.39561579952124, 13.13885481122960, 13.98076737731680, 14.70088179223446, 15.12059497497638, 16.14844729407519, 16.62868764927288, 17.49918157977795, 18.29741129905900, 19.05224813678882, 19.61684514044677

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