Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 11-s − 2·13-s + 4·17-s + 6·19-s − 8·29-s − 8·31-s + 10·37-s − 8·41-s − 2·43-s − 8·47-s − 3·49-s + 2·53-s + 12·59-s − 10·61-s + 12·67-s − 8·71-s − 6·73-s + 2·77-s − 2·79-s − 16·83-s + 14·89-s − 4·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.755·7-s + 0.301·11-s − 0.554·13-s + 0.970·17-s + 1.37·19-s − 1.48·29-s − 1.43·31-s + 1.64·37-s − 1.24·41-s − 0.304·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s + 1.56·59-s − 1.28·61-s + 1.46·67-s − 0.949·71-s − 0.702·73-s + 0.227·77-s − 0.225·79-s − 1.75·83-s + 1.48·89-s − 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(158400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{158400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 158400,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.40159646659445, −13.12622945798822, −12.58953965385784, −11.96080374938735, −11.61097549749191, −11.27639852013394, −10.79114345188352, −9.998027820693149, −9.731873106227591, −9.341757852758830, −8.698457753447735, −8.080324365216792, −7.761212654852934, −7.185575414400629, −6.928088762180400, −5.953852759702285, −5.607953978892530, −5.138459903243777, −4.641719143354022, −3.943306448800414, −3.380099672381896, −2.947640082889249, −2.009013372908785, −1.602782726551415, −0.9230875811102942, 0, 0.9230875811102942, 1.602782726551415, 2.009013372908785, 2.947640082889249, 3.380099672381896, 3.943306448800414, 4.641719143354022, 5.138459903243777, 5.607953978892530, 5.953852759702285, 6.928088762180400, 7.185575414400629, 7.761212654852934, 8.080324365216792, 8.698457753447735, 9.341757852758830, 9.731873106227591, 9.998027820693149, 10.79114345188352, 11.27639852013394, 11.61097549749191, 11.96080374938735, 12.58953965385784, 13.12622945798822, 13.40159646659445

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