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 + 2·13-s + 2·17-s + 2·19-s − 23-s − 9·27-s − 29-s − 10·31-s + 8·37-s − 6·39-s − 3·41-s + 5·43-s + 8·47-s − 6·51-s + 6·53-s − 6·57-s − 2·59-s − 9·61-s − 7·67-s + 3·69-s − 6·71-s + 10·73-s + 10·79-s + 9·81-s − 9·83-s + 3·87-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s + 0.554·13-s + 0.485·17-s + 0.458·19-s − 0.208·23-s − 1.73·27-s − 0.185·29-s − 1.79·31-s + 1.31·37-s − 0.960·39-s − 0.468·41-s + 0.762·43-s + 1.16·47-s − 0.840·51-s + 0.824·53-s − 0.794·57-s − 0.260·59-s − 1.15·61-s − 0.855·67-s + 0.361·69-s − 0.712·71-s + 1.17·73-s + 1.12·79-s + 81-s − 0.987·83-s + 0.321·87-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 \( 1 + p T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−16.24945368425078, −15.48825986977444, −15.10268187093040, −14.29697905647561, −13.68133826905678, −13.04555774564919, −12.53565010765063, −12.06115750872687, −11.55411410858469, −10.90116129796653, −10.73030040937162, −9.970761194888390, −9.390046674682306, −8.783429500250559, −7.732915010641316, −7.423333018176281, −6.649985249124424, −6.077015373273854, −5.548700980992283, −5.177563493330100, −4.241715513712944, −3.821343818692786, −2.756458903417349, −1.629707705486069, −0.9575218081380388, 0, 0.9575218081380388, 1.629707705486069, 2.756458903417349, 3.821343818692786, 4.241715513712944, 5.177563493330100, 5.548700980992283, 6.077015373273854, 6.649985249124424, 7.423333018176281, 7.732915010641316, 8.783429500250559, 9.390046674682306, 9.970761194888390, 10.73030040937162, 10.90116129796653, 11.55411410858469, 12.06115750872687, 12.53565010765063, 13.04555774564919, 13.68133826905678, 14.29697905647561, 15.10268187093040, 15.48825986977444, 16.24945368425078

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