Properties

Degree 2
Conductor $ 2^{2} \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  + 2·3-s − 5-s + 2·7-s + 9-s − 2·13-s − 2·15-s − 6·17-s + 4·21-s + 6·23-s + 25-s − 4·27-s − 6·29-s + 4·31-s − 2·35-s − 2·37-s − 4·39-s − 6·41-s − 10·43-s − 45-s − 6·47-s − 3·49-s − 12·51-s + 6·53-s − 12·59-s + 2·61-s + 2·63-s + 2·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s − 0.640·39-s − 0.937·41-s − 1.52·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s − 1.68·51-s + 0.824·53-s − 1.56·59-s + 0.256·61-s + 0.251·63-s + 0.248·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7220\)    =    \(2^{2} \cdot 5 \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{7220} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 7220,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−17.22502816595511, −17.07951414182780, −16.14690249528108, −15.32699189163018, −15.03070835956401, −14.71030098688420, −13.85250067836539, −13.43607306637841, −12.88300908329330, −12.03024050718046, −11.33813093294483, −11.02212897778077, −10.07874185154075, −9.369907460622520, −8.758614933802647, −8.347343430009299, −7.703424987420062, −7.062157231111141, −6.393475298366656, −5.140819160113988, −4.749139602334511, −3.821833994696308, −3.122347961206503, −2.318228834218007, −1.573017090111229, 0, 1.573017090111229, 2.318228834218007, 3.122347961206503, 3.821833994696308, 4.749139602334511, 5.140819160113988, 6.393475298366656, 7.062157231111141, 7.703424987420062, 8.347343430009299, 8.758614933802647, 9.369907460622520, 10.07874185154075, 11.02212897778077, 11.33813093294483, 12.03024050718046, 12.88300908329330, 13.43607306637841, 13.85250067836539, 14.71030098688420, 15.03070835956401, 15.32699189163018, 16.14690249528108, 17.07951414182780, 17.22502816595511

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