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  − 2·3-s − 5-s − 2·7-s + 9-s − 2·13-s + 2·15-s + 6·17-s + 4·19-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 − 8·57-s + 12·59-s − 2·61-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 + 0.917·19-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 − 1.05·57-s + 1.56·59-s − 0.256·61-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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
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

−19.18147473355020, −18.75494767471889, −18.05447808778205, −17.25391341605853, −16.80277001176409, −16.25495774064485, −15.79134396661238, −14.74646030906195, −14.40737581236620, −13.23663339355374, −12.68636809653418, −12.10183923269102, −11.44891369316843, −10.94937952161093, −10.01348145264732, −9.549091370160706, −8.597075432672291, −7.483399529673731, −7.120258975181624, −6.090887709202126, −5.467526097161193, −4.833319953020901, −3.611201050480833, −2.903125453540506, −1.177681937149296, 0, 1.177681937149296, 2.903125453540506, 3.611201050480833, 4.833319953020901, 5.467526097161193, 6.090887709202126, 7.120258975181624, 7.483399529673731, 8.597075432672291, 9.549091370160706, 10.01348145264732, 10.94937952161093, 11.44891369316843, 12.10183923269102, 12.68636809653418, 13.23663339355374, 14.40737581236620, 14.74646030906195, 15.79134396661238, 16.25495774064485, 16.80277001176409, 17.25391341605853, 18.05447808778205, 18.75494767471889, 19.18147473355020

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