Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 11 \cdot 43 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 0.725·3-s + 4-s + 5-s − 0.725·6-s − 1.66·7-s − 8-s − 2.47·9-s − 10-s + 11-s + 0.725·12-s − 1.34·13-s + 1.66·14-s + 0.725·15-s + 16-s − 5.15·17-s + 2.47·18-s + 1.16·19-s + 20-s − 1.20·21-s − 22-s + 7.78·23-s − 0.725·24-s + 25-s + 1.34·26-s − 3.97·27-s − 1.66·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.419·3-s + 0.5·4-s + 0.447·5-s − 0.296·6-s − 0.628·7-s − 0.353·8-s − 0.824·9-s − 0.316·10-s + 0.301·11-s + 0.209·12-s − 0.373·13-s + 0.444·14-s + 0.187·15-s + 0.250·16-s − 1.25·17-s + 0.582·18-s + 0.267·19-s + 0.223·20-s − 0.263·21-s − 0.213·22-s + 1.62·23-s − 0.148·24-s + 0.200·25-s + 0.263·26-s − 0.764·27-s − 0.314·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4730\)    =    \(2 \cdot 5 \cdot 11 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4730} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.287866938$
$L(\frac12)$  $\approx$  $1.287866938$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 - T \)
43 \( 1 - T \)
good3 \( 1 - 0.725T + 3T^{2} \)
7 \( 1 + 1.66T + 7T^{2} \)
13 \( 1 + 1.34T + 13T^{2} \)
17 \( 1 + 5.15T + 17T^{2} \)
19 \( 1 - 1.16T + 19T^{2} \)
23 \( 1 - 7.78T + 23T^{2} \)
29 \( 1 - 6.41T + 29T^{2} \)
31 \( 1 + 8.67T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + 5.51T + 41T^{2} \)
47 \( 1 + 2.09T + 47T^{2} \)
53 \( 1 - 1.29T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 1.69T + 61T^{2} \)
67 \( 1 - 0.749T + 67T^{2} \)
71 \( 1 + 5.56T + 71T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 + 0.771T + 79T^{2} \)
83 \( 1 + 4.41T + 83T^{2} \)
89 \( 1 - 7.92T + 89T^{2} \)
97 \( 1 - 5.76T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.586386618090278845600595968722, −7.61416567322618001082019646142, −6.86742590059597252942072163297, −6.34613797232929223904558473315, −5.49790521137051772539349474015, −4.62590087339270128580070910277, −3.42264525513550302244392857305, −2.76380950544971693819625588353, −1.99935709316555282938812554830, −0.66031632370940445215224059993, 0.66031632370940445215224059993, 1.99935709316555282938812554830, 2.76380950544971693819625588353, 3.42264525513550302244392857305, 4.62590087339270128580070910277, 5.49790521137051772539349474015, 6.34613797232929223904558473315, 6.86742590059597252942072163297, 7.61416567322618001082019646142, 8.586386618090278845600595968722

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