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-s − 2·9-s − 3·11-s + 6·13-s + 5·17-s − 19-s − 7·23-s + 5·27-s + 2·29-s + 5·31-s + 3·33-s − 3·37-s − 6·39-s − 2·41-s − 4·43-s + 5·47-s − 5·51-s + 53-s + 57-s − 15·59-s − 5·61-s − 9·67-s + 7·69-s − 7·73-s − 79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 0.904·11-s + 1.66·13-s + 1.21·17-s − 0.229·19-s − 1.45·23-s + 0.962·27-s + 0.371·29-s + 0.898·31-s + 0.522·33-s − 0.493·37-s − 0.960·39-s − 0.312·41-s − 0.609·43-s + 0.729·47-s − 0.700·51-s + 0.137·53-s + 0.132·57-s − 1.95·59-s − 0.640·61-s − 1.09·67-s + 0.842·69-s − 0.819·73-s − 0.112·79-s + 1/9·81-s + 1.31·83-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 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
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

−16.10454263536115, −15.50065560610272, −14.98996750315250, −14.12323072928983, −13.79888261813225, −13.37091245809283, −12.51032365915611, −12.06249520348305, −11.64025558478179, −10.91752986110805, −10.42203801221459, −10.14176000546246, −9.161828861999055, −8.576381619910657, −8.038635731659200, −7.631939652277269, −6.578326676645990, −6.043713496468321, −5.738486271491348, −5.000110615362592, −4.273270854050560, −3.401012879738710, −2.933594566952453, −1.875867525362975, −0.9958592632562323, 0, 0.9958592632562323, 1.875867525362975, 2.933594566952453, 3.401012879738710, 4.273270854050560, 5.000110615362592, 5.738486271491348, 6.043713496468321, 6.578326676645990, 7.631939652277269, 8.038635731659200, 8.576381619910657, 9.161828861999055, 10.14176000546246, 10.42203801221459, 10.91752986110805, 11.64025558478179, 12.06249520348305, 12.51032365915611, 13.37091245809283, 13.79888261813225, 14.12323072928983, 14.98996750315250, 15.50065560610272, 16.10454263536115

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