Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s − 3·9-s − 2·13-s − 2·17-s − 4·19-s − 4·23-s + 25-s − 2·29-s + 8·31-s + 4·35-s − 6·37-s + 6·41-s + 8·43-s + 3·45-s − 4·47-s + 9·49-s − 6·53-s − 4·59-s − 2·61-s + 12·63-s + 2·65-s + 8·67-s + 6·73-s + 9·81-s + 16·83-s + 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 9-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.447·45-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s − 0.256·61-s + 1.51·63-s + 0.248·65-s + 0.977·67-s + 0.702·73-s + 81-s + 1.75·83-s + 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(38720\)    =    \(2^{6} \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{38720} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 38720,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.27787429043229, −14.51413103349228, −13.98743926192081, −13.63002519236736, −12.82483891846893, −12.49732307375888, −12.11714029161833, −11.36266314009153, −10.96699432430593, −10.25415210831466, −9.849336946209800, −9.147238003860659, −8.840621701353260, −8.051876103574022, −7.680205257733274, −6.768685386478514, −6.423291104393401, −5.964428427358095, −5.222542440587419, −4.455191453946856, −3.874288246001712, −3.200997706621643, −2.654017386568744, −2.038489669302792, −0.6348491092840265, 0, 0.6348491092840265, 2.038489669302792, 2.654017386568744, 3.200997706621643, 3.874288246001712, 4.455191453946856, 5.222542440587419, 5.964428427358095, 6.423291104393401, 6.768685386478514, 7.680205257733274, 8.051876103574022, 8.840621701353260, 9.147238003860659, 9.849336946209800, 10.25415210831466, 10.96699432430593, 11.36266314009153, 12.11714029161833, 12.49732307375888, 12.82483891846893, 13.63002519236736, 13.98743926192081, 14.51413103349228, 15.27787429043229

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