Properties

Degree 2
Conductor $ 2^{2} \cdot 5 \cdot 11^{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 − 5-s − 7-s − 2·9-s + 2·13-s − 15-s + 2·19-s − 21-s + 25-s − 5·27-s − 6·29-s − 4·31-s + 35-s − 4·37-s + 2·39-s + 9·41-s − 43-s + 2·45-s − 3·47-s − 6·49-s − 6·53-s + 2·57-s − 61-s + 2·63-s − 2·65-s − 13·67-s − 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.554·13-s − 0.258·15-s + 0.458·19-s − 0.218·21-s + 1/5·25-s − 0.962·27-s − 1.11·29-s − 0.718·31-s + 0.169·35-s − 0.657·37-s + 0.320·39-s + 1.40·41-s − 0.152·43-s + 0.298·45-s − 0.437·47-s − 6/7·49-s − 0.824·53-s + 0.264·57-s − 0.128·61-s + 0.251·63-s − 0.248·65-s − 1.58·67-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2420\)    =    \(2^{2} \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2420} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 2420,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−19.26729312581519, −19.00544779251722, −18.03738904992241, −17.56149344286069, −16.57948797311697, −16.21086733453484, −15.46713886796489, −14.73608052438199, −14.31839153786630, −13.46107158933369, −12.98381701489617, −12.13455288038144, −11.38149554846219, −10.89846260980766, −9.917050322894070, −9.134787141450749, −8.681970147137822, −7.786672654079837, −7.263525602150825, −6.163201879082631, −5.545542950761597, −4.406734484955620, −3.485126869510987, −2.915251327956251, −1.648138373382446, 0, 1.648138373382446, 2.915251327956251, 3.485126869510987, 4.406734484955620, 5.545542950761597, 6.163201879082631, 7.263525602150825, 7.786672654079837, 8.681970147137822, 9.134787141450749, 9.917050322894070, 10.89846260980766, 11.38149554846219, 12.13455288038144, 12.98381701489617, 13.46107158933369, 14.31839153786630, 14.73608052438199, 15.46713886796489, 16.21086733453484, 16.57948797311697, 17.56149344286069, 18.03738904992241, 19.00544779251722, 19.26729312581519

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