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  + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 7-s + 9-s − 2·10-s + 4·11-s + 2·12-s − 13-s + 2·14-s − 15-s − 4·16-s − 5·17-s + 2·18-s − 6·19-s − 2·20-s + 21-s + 8·22-s + 25-s − 2·26-s + 27-s + 2·28-s + 9·29-s − 2·30-s + 4·31-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.577·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s − 16-s − 1.21·17-s + 0.471·18-s − 1.37·19-s − 0.447·20-s + 0.218·21-s + 1.70·22-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.377·28-s + 1.67·29-s − 0.365·30-s + 0.718·31-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  =  $(2,\ 55545,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.824130676$
$L(\frac12)$  $\approx$  $5.824130676$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−14.48271843457153, −13.91843249127766, −13.39453039552231, −13.04292520329490, −12.42277109285067, −11.97519886276307, −11.54622567859148, −11.10172050472719, −10.37511251330776, −9.835460224127161, −8.962351413983665, −8.703647014281345, −8.269948195753581, −7.399775808657748, −6.819741264537615, −6.359012124834661, −6.032852246908801, −4.855681233256496, −4.629985205767082, −4.209189032845916, −3.619332848259139, −2.949748807610243, −2.332412030378960, −1.717892286586642, −0.6121549816119179, 0.6121549816119179, 1.717892286586642, 2.332412030378960, 2.949748807610243, 3.619332848259139, 4.209189032845916, 4.629985205767082, 4.855681233256496, 6.032852246908801, 6.359012124834661, 6.819741264537615, 7.399775808657748, 8.269948195753581, 8.703647014281345, 8.962351413983665, 9.835460224127161, 10.37511251330776, 11.10172050472719, 11.54622567859148, 11.97519886276307, 12.42277109285067, 13.04292520329490, 13.39453039552231, 13.91843249127766, 14.48271843457153

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