Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 11-s + 4·13-s + 6·19-s + 3·23-s + 4·27-s − 3·29-s + 2·33-s − 9·37-s − 8·39-s − 2·41-s + 9·43-s + 6·47-s − 6·53-s − 12·57-s + 8·59-s + 10·61-s − 67-s − 6·69-s + 7·71-s − 2·73-s + 9·79-s − 11·81-s − 12·83-s + 6·87-s + 4·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 1.37·19-s + 0.625·23-s + 0.769·27-s − 0.557·29-s + 0.348·33-s − 1.47·37-s − 1.28·39-s − 0.312·41-s + 1.37·43-s + 0.875·47-s − 0.824·53-s − 1.58·57-s + 1.04·59-s + 1.28·61-s − 0.122·67-s − 0.722·69-s + 0.830·71-s − 0.234·73-s + 1.01·79-s − 1.22·81-s − 1.31·83-s + 0.643·87-s + 0.423·89-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  =  0
Selberg data  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.453028750$
$L(\frac12)$  $\approx$  $1.453028750$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−15.65003020768915, −15.57977343407912, −14.41475067930450, −14.12868655561987, −13.41426392620584, −12.89138000607752, −12.33018084350634, −11.68703131761063, −11.32573826463076, −10.80674787539644, −10.31251843512590, −9.643090984909586, −8.882278303870301, −8.488378124970329, −7.530079288724180, −7.138573515976322, −6.339569095476486, −5.851671838484089, −5.260131141219358, −4.856397392216040, −3.808097815658689, −3.309775839477128, −2.328545925914458, −1.255981859463048, −0.6058182960215041, 0.6058182960215041, 1.255981859463048, 2.328545925914458, 3.309775839477128, 3.808097815658689, 4.856397392216040, 5.260131141219358, 5.851671838484089, 6.339569095476486, 7.138573515976322, 7.530079288724180, 8.488378124970329, 8.882278303870301, 9.643090984909586, 10.31251843512590, 10.80674787539644, 11.32573826463076, 11.68703131761063, 12.33018084350634, 12.89138000607752, 13.41426392620584, 14.12868655561987, 14.41475067930450, 15.57977343407912, 15.65003020768915

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