Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 47^{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 − 6-s + 2·7-s − 8-s + 9-s + 11-s + 12-s + 4·13-s − 2·14-s + 16-s − 6·17-s − 18-s + 4·19-s + 2·21-s − 22-s − 6·23-s − 24-s − 5·25-s − 4·26-s + 27-s + 2·28-s − 6·29-s − 8·31-s − 32-s + 33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.436·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s − 25-s − 0.784·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145794 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145794 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(145794\)    =    \(2 \cdot 3 \cdot 11 \cdot 47^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{145794} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 145794,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.368377144$
$L(\frac12)$  $\approx$  $1.368377144$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;47\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;47\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
11 \( 1 - T \)
47 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.31481748099932, −13.11680049076437, −12.29376049696711, −11.75203198908572, −11.37542338433041, −11.04731424510187, −10.45271780012179, −9.908188076223682, −9.467398821697591, −8.869737330212969, −8.620515228529207, −8.078888991510821, −7.693316624088635, −7.050573612568338, −6.659582149870925, −6.037700720416738, −5.395000724644865, −4.964427509629190, −3.987361254745587, −3.755371357948488, −3.221048268896768, −2.152825712677233, −1.837463479989450, −1.478201237538283, −0.3529693682283094, 0.3529693682283094, 1.478201237538283, 1.837463479989450, 2.152825712677233, 3.221048268896768, 3.755371357948488, 3.987361254745587, 4.964427509629190, 5.395000724644865, 6.037700720416738, 6.659582149870925, 7.050573612568338, 7.693316624088635, 8.078888991510821, 8.620515228529207, 8.869737330212969, 9.467398821697591, 9.908188076223682, 10.45271780012179, 11.04731424510187, 11.37542338433041, 11.75203198908572, 12.29376049696711, 13.11680049076437, 13.31481748099932

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