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 + 2·11-s − 4·17-s − 6·19-s + 3·23-s − 9·27-s + 9·29-s − 4·31-s − 6·33-s + 4·37-s + 7·41-s − 5·43-s − 8·47-s + 12·51-s + 2·53-s + 18·57-s + 10·59-s − 61-s − 9·67-s − 9·69-s − 2·71-s − 4·73-s − 10·79-s + 9·81-s + 7·83-s − 27·87-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s + 0.603·11-s − 0.970·17-s − 1.37·19-s + 0.625·23-s − 1.73·27-s + 1.67·29-s − 0.718·31-s − 1.04·33-s + 0.657·37-s + 1.09·41-s − 0.762·43-s − 1.16·47-s + 1.68·51-s + 0.274·53-s + 2.38·57-s + 1.30·59-s − 0.128·61-s − 1.09·67-s − 1.08·69-s − 0.237·71-s − 0.468·73-s − 1.12·79-s + 81-s + 0.768·83-s − 2.89·87-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.12555979460961, −15.61087900161358, −14.86160364425339, −14.54583442661604, −13.56933834075725, −12.96540521667254, −12.71600307614288, −11.94016438482393, −11.55918858249818, −11.06161639108464, −10.52366100671689, −10.12499345645199, −9.294232895721718, −8.729595374054972, −8.031777453811011, −7.040224344905625, −6.735844560309405, −6.189110704144848, −5.697310316276505, −4.732809409354092, −4.564242558785203, −3.796348836644142, −2.671136584145698, −1.733161434170461, −0.8866066671544230, 0, 0.8866066671544230, 1.733161434170461, 2.671136584145698, 3.796348836644142, 4.564242558785203, 4.732809409354092, 5.697310316276505, 6.189110704144848, 6.735844560309405, 7.040224344905625, 8.031777453811011, 8.729595374054972, 9.294232895721718, 10.12499345645199, 10.52366100671689, 11.06161639108464, 11.55918858249818, 11.94016438482393, 12.71600307614288, 12.96540521667254, 13.56933834075725, 14.54583442661604, 14.86160364425339, 15.61087900161358, 16.12555979460961

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