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 − 5·11-s + 8·17-s − 2·19-s − 7·23-s + 4·27-s − 3·29-s + 4·31-s + 10·33-s + 37-s + 2·41-s + 3·43-s − 6·47-s − 16·51-s − 10·53-s + 4·57-s − 4·59-s + 6·61-s + 13·67-s + 14·69-s − 5·71-s + 6·73-s + 13·79-s − 11·81-s − 16·83-s + 6·87-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.50·11-s + 1.94·17-s − 0.458·19-s − 1.45·23-s + 0.769·27-s − 0.557·29-s + 0.718·31-s + 1.74·33-s + 0.164·37-s + 0.312·41-s + 0.457·43-s − 0.875·47-s − 2.24·51-s − 1.37·53-s + 0.529·57-s − 0.520·59-s + 0.768·61-s + 1.58·67-s + 1.68·69-s − 0.593·71-s + 0.702·73-s + 1.46·79-s − 1.22·81-s − 1.75·83-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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−15.99256016353652, −15.70058513966605, −14.87600348467512, −14.28405436115746, −13.85277080963304, −13.01633432593897, −12.53690000432490, −12.19424687136117, −11.52090741391700, −11.00258225013793, −10.47031487850916, −9.937901656129093, −9.581588202188071, −8.376347269716846, −8.027800180850676, −7.534964613529119, −6.669564044290661, −5.986412067182705, −5.604623810289834, −5.088070774282148, −4.414255839793834, −3.495026834506201, −2.792159088262093, −1.907552043113546, −0.8361559572439254, 0, 0.8361559572439254, 1.907552043113546, 2.792159088262093, 3.495026834506201, 4.414255839793834, 5.088070774282148, 5.604623810289834, 5.986412067182705, 6.669564044290661, 7.534964613529119, 8.027800180850676, 8.376347269716846, 9.581588202188071, 9.937901656129093, 10.47031487850916, 11.00258225013793, 11.52090741391700, 12.19424687136117, 12.53690000432490, 13.01633432593897, 13.85277080963304, 14.28405436115746, 14.87600348467512, 15.70058513966605, 15.99256016353652

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