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-s + 3-s − 4-s + 5-s + 6-s − 7-s − 3·8-s + 9-s + 10-s + 4·11-s − 12-s + 2·13-s − 14-s + 15-s − 16-s − 6·17-s + 18-s + 4·19-s − 20-s − 21-s + 4·22-s − 3·24-s + 25-s + 2·26-s + 27-s + 28-s + 6·29-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 + 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.852·22-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-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  =  $(2,\ 55545,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.427257109$
$L(\frac12)$  $\approx$  $4.427257109$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−14.02321687314351, −13.95519183180598, −13.57492015079286, −13.06772703819628, −12.44526139225408, −12.02934890184119, −11.50731589940646, −10.87131153923201, −10.06487810006496, −9.752085811102779, −9.150139812318144, −8.659730618712150, −8.505965886364467, −7.539581302139163, −6.744612037389992, −6.456021172378383, −5.960885711741041, −5.147784510626677, −4.619291194884922, −4.058573925244711, −3.568887735161823, −2.910954135984362, −2.338558745129176, −1.367673976972982, −0.6678235998910487, 0.6678235998910487, 1.367673976972982, 2.338558745129176, 2.910954135984362, 3.568887735161823, 4.058573925244711, 4.619291194884922, 5.147784510626677, 5.960885711741041, 6.456021172378383, 6.744612037389992, 7.539581302139163, 8.505965886364467, 8.659730618712150, 9.150139812318144, 9.752085811102779, 10.06487810006496, 10.87131153923201, 11.50731589940646, 12.02934890184119, 12.44526139225408, 13.06772703819628, 13.57492015079286, 13.95519183180598, 14.02321687314351

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