Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 23^{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 − 2·4-s + 5-s + 7-s + 9-s − 11-s − 2·12-s + 15-s + 4·16-s − 2·17-s + 3·19-s − 2·20-s + 21-s + 25-s + 27-s − 2·28-s − 6·29-s − 33-s + 35-s − 2·36-s − 6·37-s − 3·41-s + 6·43-s + 2·44-s + 45-s − 7·47-s + 4·48-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s + 0.258·15-s + 16-s − 0.485·17-s + 0.688·19-s − 0.447·20-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 1.11·29-s − 0.174·33-s + 0.169·35-s − 1/3·36-s − 0.986·37-s − 0.468·41-s + 0.914·43-s + 0.301·44-s + 0.149·45-s − 1.02·47-s + 0.577·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(55545\)    =    \(3 \cdot 5 \cdot 7 \cdot 23^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{55545} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 55545,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−14.61789120480066, −14.08815519719772, −13.56469456032664, −13.36301134984621, −12.75714618576414, −12.27887659012128, −11.62449269063173, −10.98874354321455, −10.42001317780360, −9.933174959276428, −9.411293344199375, −8.963699990194170, −8.553586160277826, −7.912376365658849, −7.483434300593919, −6.848355593541764, −6.065700791693935, −5.435863607123855, −5.014894896953897, −4.430234132875259, −3.715032233372159, −3.272973128864865, −2.405570082787162, −1.762007809569565, −0.9941001589179868, 0, 0.9941001589179868, 1.762007809569565, 2.405570082787162, 3.272973128864865, 3.715032233372159, 4.430234132875259, 5.014894896953897, 5.435863607123855, 6.065700791693935, 6.848355593541764, 7.483434300593919, 7.912376365658849, 8.553586160277826, 8.963699990194170, 9.411293344199375, 9.933174959276428, 10.42001317780360, 10.98874354321455, 11.62449269063173, 12.27887659012128, 12.75714618576414, 13.36301134984621, 13.56469456032664, 14.08815519719772, 14.61789120480066

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