Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} \cdot 7^{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  − 3·3-s + 6·9-s − 11-s − 4·13-s − 5·17-s + 19-s + 2·23-s − 9·27-s − 8·29-s + 10·31-s + 3·33-s − 6·37-s + 12·39-s + 3·41-s − 4·43-s + 4·47-s + 15·51-s + 6·53-s − 3·57-s + 8·59-s − 10·61-s + 67-s − 6·69-s + 12·71-s − 3·73-s − 6·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s − 0.301·11-s − 1.10·13-s − 1.21·17-s + 0.229·19-s + 0.417·23-s − 1.73·27-s − 1.48·29-s + 1.79·31-s + 0.522·33-s − 0.986·37-s + 1.92·39-s + 0.468·41-s − 0.609·43-s + 0.583·47-s + 2.10·51-s + 0.824·53-s − 0.397·57-s + 1.04·59-s − 1.28·61-s + 0.122·67-s − 0.722·69-s + 1.42·71-s − 0.351·73-s − 0.675·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{19600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 19600,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−15.90657016377376, −15.59070679571692, −15.11623418575742, −14.36029829001779, −13.58394444797434, −13.07563686496996, −12.58507550283419, −11.98090518113056, −11.56980921626342, −11.10500701516798, −10.43682042337769, −10.09557648491162, −9.408806232298833, −8.737732057725102, −7.873932523243992, −7.094425888212846, −6.907804836361721, −6.073751739808229, −5.589965103349775, −4.847192388206211, −4.611971891723123, −3.720250276678339, −2.626674404670953, −1.856976039102084, −0.7770269505093630, 0, 0.7770269505093630, 1.856976039102084, 2.626674404670953, 3.720250276678339, 4.611971891723123, 4.847192388206211, 5.589965103349775, 6.073751739808229, 6.907804836361721, 7.094425888212846, 7.873932523243992, 8.737732057725102, 9.408806232298833, 10.09557648491162, 10.43682042337769, 11.10500701516798, 11.56980921626342, 11.98090518113056, 12.58507550283419, 13.07563686496996, 13.58394444797434, 14.36029829001779, 15.11623418575742, 15.59070679571692, 15.90657016377376

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