Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 23^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 5-s − 7-s + 9-s − 3·11-s − 2·12-s − 4·13-s + 15-s + 4·16-s + 6·17-s + 19-s − 2·20-s − 21-s + 25-s + 27-s + 2·28-s + 6·29-s + 8·31-s − 3·33-s − 35-s − 2·36-s + 10·37-s − 4·39-s − 3·41-s + 10·43-s + 6·44-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s − 1.10·13-s + 0.258·15-s + 16-s + 1.45·17-s + 0.229·19-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 + 1.43·31-s − 0.522·33-s − 0.169·35-s − 1/3·36-s + 1.64·37-s − 0.640·39-s − 0.468·41-s + 1.52·43-s + 0.904·44-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  =  0
Selberg data  =  $(2,\ 55545,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.510070243$
$L(\frac12)$  $\approx$  $2.510070243$
$L(\frac{3}{2})$   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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.18808549027958, −13.91191904954432, −13.56133938994077, −12.84138224841519, −12.50800178140907, −12.15139758977923, −11.38787043188240, −10.37760057587656, −10.21269781627352, −9.746412831641297, −9.356127223623477, −8.735000735493313, −8.034133482968831, −7.800778098098589, −7.246297027219910, −6.343281676614085, −5.849795789376670, −5.132881975746678, −4.789973331738052, −4.140799537005224, −3.349366969059214, −2.777642132236909, −2.371566289401327, −1.159816296433942, −0.6093847851134433, 0.6093847851134433, 1.159816296433942, 2.371566289401327, 2.777642132236909, 3.349366969059214, 4.140799537005224, 4.789973331738052, 5.132881975746678, 5.849795789376670, 6.343281676614085, 7.246297027219910, 7.800778098098589, 8.034133482968831, 8.735000735493313, 9.356127223623477, 9.746412831641297, 10.21269781627352, 10.37760057587656, 11.38787043188240, 12.15139758977923, 12.50800178140907, 12.84138224841519, 13.56133938994077, 13.91191904954432, 14.18808549027958

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