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 + 4·11-s + 2·13-s + 8·17-s − 6·19-s − 4·23-s + 4·27-s − 6·29-s + 4·31-s − 8·33-s + 10·37-s − 4·39-s − 4·41-s + 4·43-s − 4·47-s − 16·51-s − 10·53-s + 12·57-s − 14·59-s + 10·61-s − 4·67-s + 8·69-s − 12·71-s + 4·73-s − 4·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 1.94·17-s − 1.37·19-s − 0.834·23-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 1.39·33-s + 1.64·37-s − 0.640·39-s − 0.624·41-s + 0.609·43-s − 0.583·47-s − 2.24·51-s − 1.37·53-s + 1.58·57-s − 1.82·59-s + 1.28·61-s − 0.488·67-s + 0.963·69-s − 1.42·71-s + 0.468·73-s − 0.450·79-s − 1.22·81-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−16.25921563421121, −15.47794315771135, −14.70261549407519, −14.48875489227833, −13.85241014522374, −13.05971109261885, −12.50632413671496, −12.06477857784050, −11.56064150923614, −11.09632818238921, −10.53536069873297, −9.867922604549110, −9.426481562076698, −8.597692611447920, −8.046964884567384, −7.401461283955993, −6.536461254094281, −6.057741830791649, −5.848820649232362, −4.934580224059062, −4.260157263429552, −3.674403782934711, −2.842006653410816, −1.661013161275982, −1.058089475677375, 0, 1.058089475677375, 1.661013161275982, 2.842006653410816, 3.674403782934711, 4.260157263429552, 4.934580224059062, 5.848820649232362, 6.057741830791649, 6.536461254094281, 7.401461283955993, 8.046964884567384, 8.597692611447920, 9.426481562076698, 9.867922604549110, 10.53536069873297, 11.09632818238921, 11.56064150923614, 12.06477857784050, 12.50632413671496, 13.05971109261885, 13.85241014522374, 14.48875489227833, 14.70261549407519, 15.47794315771135, 16.25921563421121

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