Properties

Degree 2
Conductor $ 2^{4} \cdot 5 \cdot 19^{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  + 5-s + 4·7-s − 3·9-s − 4·11-s + 2·13-s + 2·17-s − 4·23-s + 25-s + 2·29-s − 8·31-s + 4·35-s − 6·37-s + 6·41-s + 8·43-s − 3·45-s − 4·47-s + 9·49-s − 6·53-s − 4·55-s − 4·59-s − 2·61-s − 12·63-s + 2·65-s + 8·67-s − 6·73-s − 16·77-s + 9·81-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s − 0.447·45-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.539·55-s − 0.520·59-s − 0.256·61-s − 1.51·63-s + 0.248·65-s + 0.977·67-s − 0.702·73-s − 1.82·77-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(28880\)    =    \(2^{4} \cdot 5 \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{28880} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 28880,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
19 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.45823784602950, −14.60933351069785, −14.49745806667775, −13.93477321363045, −13.49333869886481, −12.75013720607818, −12.28358819154374, −11.55796571430909, −11.17350730890962, −10.58208939177142, −10.35363033826045, −9.304527836590987, −8.940484557015429, −8.212635617784777, −7.842101020430426, −7.462126726003729, −6.373993442438209, −5.868303915341730, −5.203245480697315, −5.024299131204806, −4.051300100832334, −3.320722805456653, −2.476923182060125, −1.969928912102810, −1.138474612852257, 0, 1.138474612852257, 1.969928912102810, 2.476923182060125, 3.320722805456653, 4.051300100832334, 5.024299131204806, 5.203245480697315, 5.868303915341730, 6.373993442438209, 7.462126726003729, 7.842101020430426, 8.212635617784777, 8.940484557015429, 9.304527836590987, 10.35363033826045, 10.58208939177142, 11.17350730890962, 11.55796571430909, 12.28358819154374, 12.75013720607818, 13.49333869886481, 13.93477321363045, 14.49745806667775, 14.60933351069785, 15.45823784602950

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