Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7^{2} $
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 + 6·13-s − 15-s − 2·17-s − 8·19-s − 8·23-s + 25-s + 27-s − 2·29-s + 4·31-s − 2·37-s + 6·39-s + 6·41-s − 4·43-s − 45-s + 8·47-s − 2·51-s + 10·53-s − 8·57-s + 4·59-s + 2·61-s − 6·65-s − 4·67-s − 8·69-s + 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.66·13-s − 0.258·15-s − 0.485·17-s − 1.83·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.328·37-s + 0.960·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 0.280·51-s + 1.37·53-s − 1.05·57-s + 0.520·59-s + 0.256·61-s − 0.744·65-s − 0.488·67-s − 0.963·69-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11760\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{11760} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 11760,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.216177094$
$L(\frac12)$  $\approx$  $2.216177094$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−16.20064879929378, −15.76796683111140, −15.36487231904657, −14.73434781127721, −14.10405189580013, −13.58630998785843, −13.03806942009920, −12.53648975776441, −11.74709936763120, −11.25905975835320, −10.46684847898243, −10.25010337733072, −9.166342448850215, −8.679898149267283, −8.280506737762416, −7.693549770934459, −6.800582015975715, −6.254767463202724, −5.685364146536516, −4.498240300928309, −4.017863691081947, −3.550297980436445, −2.436822780531490, −1.839821954639829, −0.6534134344256401, 0.6534134344256401, 1.839821954639829, 2.436822780531490, 3.550297980436445, 4.017863691081947, 4.498240300928309, 5.685364146536516, 6.254767463202724, 6.800582015975715, 7.693549770934459, 8.280506737762416, 8.679898149267283, 9.166342448850215, 10.25010337733072, 10.46684847898243, 11.25905975835320, 11.74709936763120, 12.53648975776441, 13.03806942009920, 13.58630998785843, 14.10405189580013, 14.73434781127721, 15.36487231904657, 15.76796683111140, 16.20064879929378

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