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  − 2·3-s + 9-s + 11-s + 2·13-s − 4·17-s + 5·23-s + 4·27-s − 3·29-s + 10·31-s − 2·33-s − 5·37-s − 4·39-s − 10·41-s − 5·43-s − 4·47-s + 8·51-s − 10·53-s + 10·59-s + 10·61-s + 5·67-s − 10·69-s − 3·71-s + 10·73-s − 13·79-s − 11·81-s − 10·83-s + 6·87-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.970·17-s + 1.04·23-s + 0.769·27-s − 0.557·29-s + 1.79·31-s − 0.348·33-s − 0.821·37-s − 0.640·39-s − 1.56·41-s − 0.762·43-s − 0.583·47-s + 1.12·51-s − 1.37·53-s + 1.30·59-s + 1.28·61-s + 0.610·67-s − 1.20·69-s − 0.356·71-s + 1.17·73-s − 1.46·79-s − 1.22·81-s − 1.09·83-s + 0.643·87-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.93268768545204, −15.55915294973212, −15.02280150919149, −14.26459092331110, −13.75579619870684, −13.04728508311949, −12.76641689683410, −11.85995883339210, −11.51009832393516, −11.21098692186627, −10.44458446951449, −10.05333553726797, −9.269645009654721, −8.517700759483474, −8.284418119907593, −7.153973012061621, −6.584250184413103, −6.402663757893261, −5.444368267553638, −5.035276513726638, −4.400067335224099, −3.560505189219432, −2.817947078674961, −1.787643027074978, −0.9455357247611698, 0, 0.9455357247611698, 1.787643027074978, 2.817947078674961, 3.560505189219432, 4.400067335224099, 5.035276513726638, 5.444368267553638, 6.402663757893261, 6.584250184413103, 7.153973012061621, 8.284418119907593, 8.517700759483474, 9.269645009654721, 10.05333553726797, 10.44458446951449, 11.21098692186627, 11.51009832393516, 11.85995883339210, 12.76641689683410, 13.04728508311949, 13.75579619870684, 14.26459092331110, 15.02280150919149, 15.55915294973212, 15.93268768545204

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