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 − 2·13-s + 3·17-s + 8·19-s + 9·23-s + 4·27-s − 6·29-s + 5·31-s + 8·37-s + 4·39-s + 3·41-s + 10·43-s − 3·47-s − 6·51-s + 6·53-s − 16·57-s + 12·59-s + 4·61-s − 2·67-s − 18·69-s + 9·71-s + 10·73-s − 5·79-s − 11·81-s − 6·83-s + 12·87-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 0.554·13-s + 0.727·17-s + 1.83·19-s + 1.87·23-s + 0.769·27-s − 1.11·29-s + 0.898·31-s + 1.31·37-s + 0.640·39-s + 0.468·41-s + 1.52·43-s − 0.437·47-s − 0.840·51-s + 0.824·53-s − 2.11·57-s + 1.56·59-s + 0.512·61-s − 0.244·67-s − 2.16·69-s + 1.06·71-s + 1.17·73-s − 0.562·79-s − 1.22·81-s − 0.658·83-s + 1.28·87-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  =  0
Selberg data  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.741545377$
$L(\frac12)$  $\approx$  $1.741545377$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−15.85475191722385, −15.18986818059404, −14.55526286743464, −14.18149188427859, −13.35602660653839, −12.87626138465760, −12.28699224589531, −11.76906209995307, −11.25218724873558, −10.96798679861375, −10.10775106957859, −9.628309657057280, −9.150381128600005, −8.281939727731991, −7.531240807912615, −7.153778110731776, −6.476195461265707, −5.612314153760234, −5.406229046443314, −4.802713301618459, −3.963076987909330, −3.075503749211923, −2.498745903941090, −1.102568226497828, −0.7335407485896233, 0.7335407485896233, 1.102568226497828, 2.498745903941090, 3.075503749211923, 3.963076987909330, 4.802713301618459, 5.406229046443314, 5.612314153760234, 6.476195461265707, 7.153778110731776, 7.531240807912615, 8.281939727731991, 9.150381128600005, 9.628309657057280, 10.10775106957859, 10.96798679861375, 11.25218724873558, 11.76906209995307, 12.28699224589531, 12.87626138465760, 13.35602660653839, 14.18149188427859, 14.55526286743464, 15.18986818059404, 15.85475191722385

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