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-s − 2·9-s − 2·13-s − 2·17-s − 6·19-s − 3·23-s + 5·27-s + 7·29-s − 2·31-s + 8·37-s + 2·39-s + 5·41-s + 7·43-s + 2·51-s − 6·53-s + 6·57-s − 10·59-s + 7·61-s − 5·67-s + 3·69-s + 2·71-s + 6·73-s + 2·79-s + 81-s − 11·83-s − 7·87-s + 9·89-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 0.554·13-s − 0.485·17-s − 1.37·19-s − 0.625·23-s + 0.962·27-s + 1.29·29-s − 0.359·31-s + 1.31·37-s + 0.320·39-s + 0.780·41-s + 1.06·43-s + 0.280·51-s − 0.824·53-s + 0.794·57-s − 1.30·59-s + 0.896·61-s − 0.610·67-s + 0.361·69-s + 0.237·71-s + 0.702·73-s + 0.225·79-s + 1/9·81-s − 1.20·83-s − 0.750·87-s + 0.953·89-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  =  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 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−16.01350571330217, −15.48687171944049, −14.80173435417502, −14.32765334454107, −13.93252105085251, −13.08449921179947, −12.56898175889475, −12.21367393975299, −11.38864308532341, −11.14110296821481, −10.44686051009649, −9.993664464354954, −9.136001695421174, −8.743009213736010, −8.010504770756478, −7.508803765073249, −6.592316768360841, −6.185010049180910, −5.723294948212976, −4.664283472254201, −4.549742904482632, −3.526722959572696, −2.608293285227328, −2.140390783843939, −0.8785497492497914, 0, 0.8785497492497914, 2.140390783843939, 2.608293285227328, 3.526722959572696, 4.549742904482632, 4.664283472254201, 5.723294948212976, 6.185010049180910, 6.592316768360841, 7.508803765073249, 8.010504770756478, 8.743009213736010, 9.136001695421174, 9.993664464354954, 10.44686051009649, 11.14110296821481, 11.38864308532341, 12.21367393975299, 12.56898175889475, 13.08449921179947, 13.93252105085251, 14.32765334454107, 14.80173435417502, 15.48687171944049, 16.01350571330217

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