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  − 2-s + 3-s − 4-s + 5-s − 6-s + 7-s + 3·8-s + 9-s − 10-s − 5·11-s − 12-s + 13-s − 14-s + 15-s − 16-s − 18-s − 5·19-s − 20-s + 21-s + 5·22-s + 3·24-s + 25-s − 26-s + 27-s − 28-s − 6·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.235·18-s − 1.14·19-s − 0.223·20-s + 0.218·21-s + 1.06·22-s + 0.612·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-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 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
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

−14.61709414614128, −14.13506469778473, −13.58957574408894, −13.11490659518968, −12.87066789694333, −12.32477168321812, −11.28555479432386, −10.88373577532710, −10.47467446192710, −9.962334706284970, −9.446473039084364, −8.934342087093279, −8.358109215293577, −8.150509698694166, −7.332997001879334, −7.199278247119552, −6.054187254625461, −5.611174667286785, −4.989598144401948, −4.339122643111985, −3.891337601627931, −2.950960648167658, −2.279310509495494, −1.797029222364884, −0.8758596679396492, 0, 0.8758596679396492, 1.797029222364884, 2.279310509495494, 2.950960648167658, 3.891337601627931, 4.339122643111985, 4.989598144401948, 5.611174667286785, 6.054187254625461, 7.199278247119552, 7.332997001879334, 8.150509698694166, 8.358109215293577, 8.934342087093279, 9.446473039084364, 9.962334706284970, 10.47467446192710, 10.88373577532710, 11.28555479432386, 12.32477168321812, 12.87066789694333, 13.11490659518968, 13.58957574408894, 14.13506469778473, 14.61709414614128

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