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 − 3·11-s + 4·13-s + 2·19-s + 3·23-s + 4·27-s + 9·29-s + 8·31-s + 6·33-s + 5·37-s − 8·39-s + 6·41-s − 11·43-s + 6·47-s + 6·53-s − 4·57-s + 10·61-s − 5·67-s − 6·69-s − 15·71-s + 10·73-s + 7·79-s − 11·81-s + 12·83-s − 18·87-s + 12·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 0.904·11-s + 1.10·13-s + 0.458·19-s + 0.625·23-s + 0.769·27-s + 1.67·29-s + 1.43·31-s + 1.04·33-s + 0.821·37-s − 1.28·39-s + 0.937·41-s − 1.67·43-s + 0.875·47-s + 0.824·53-s − 0.529·57-s + 1.28·61-s − 0.610·67-s − 0.722·69-s − 1.78·71-s + 1.17·73-s + 0.787·79-s − 1.22·81-s + 1.31·83-s − 1.92·87-s + 1.27·89-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  =  0
Selberg data  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.537473439$
$L(\frac12)$  $\approx$  $1.537473439$
$L(\frac{3}{2})$   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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−15.89309198361505, −15.25712257933645, −14.73404260783065, −13.75699609398024, −13.58500136769743, −12.94295545092337, −12.17156595303463, −11.90184079583552, −11.20704523507744, −10.80163723504998, −10.26578711382017, −9.781782783061883, −8.816721847264249, −8.380324488112646, −7.771600458235001, −6.941792480478141, −6.358457332402285, −5.952865413343144, −5.181388327180603, −4.811733983942328, −4.004446569661878, −3.052237541416124, −2.497212080444745, −1.167851349895466, −0.6529069534363613, 0.6529069534363613, 1.167851349895466, 2.497212080444745, 3.052237541416124, 4.004446569661878, 4.811733983942328, 5.181388327180603, 5.952865413343144, 6.358457332402285, 6.941792480478141, 7.771600458235001, 8.380324488112646, 8.816721847264249, 9.781782783061883, 10.26578711382017, 10.80163723504998, 11.20704523507744, 11.90184079583552, 12.17156595303463, 12.94295545092337, 13.58500136769743, 13.75699609398024, 14.73404260783065, 15.25712257933645, 15.89309198361505

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