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 − 2.56·3-s + 4-s + 5-s − 2.56·6-s + 3.71·7-s + 8-s + 3.56·9-s + 10-s − 11-s − 2.56·12-s + 5.57·13-s + 3.71·14-s − 2.56·15-s + 16-s + 3.56·17-s + 3.56·18-s + 7.50·19-s + 20-s − 9.51·21-s − 22-s − 3.57·23-s − 2.56·24-s + 25-s + 5.57·26-s − 1.45·27-s + 3.71·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.47·3-s + 0.5·4-s + 0.447·5-s − 1.04·6-s + 1.40·7-s + 0.353·8-s + 1.18·9-s + 0.316·10-s − 0.301·11-s − 0.739·12-s + 1.54·13-s + 0.992·14-s − 0.661·15-s + 0.250·16-s + 0.865·17-s + 0.840·18-s + 1.72·19-s + 0.223·20-s − 2.07·21-s − 0.213·22-s − 0.746·23-s − 0.523·24-s + 0.200·25-s + 1.09·26-s − 0.279·27-s + 0.701·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.917064083$
$L(\frac12)$  $\approx$  $2.917064083$
$L(\frac{3}{2})$   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 \( 1 + 2.56T + 3T^{2} \)
7 \( 1 - 3.71T + 7T^{2} \)
13 \( 1 - 5.57T + 13T^{2} \)
17 \( 1 - 3.56T + 17T^{2} \)
19 \( 1 - 7.50T + 19T^{2} \)
23 \( 1 + 3.57T + 23T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 + 8.61T + 31T^{2} \)
37 \( 1 + 1.18T + 37T^{2} \)
41 \( 1 - 9.18T + 41T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 + 9.37T + 53T^{2} \)
59 \( 1 + 4.51T + 59T^{2} \)
61 \( 1 + 10.7T + 61T^{2} \)
67 \( 1 - 6.61T + 67T^{2} \)
71 \( 1 + 0.469T + 71T^{2} \)
73 \( 1 + 1.64T + 73T^{2} \)
79 \( 1 + 4.43T + 79T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 - 0.511T + 97T^{2} \)
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

−7.956025303016115457145403627699, −7.57156544957932290468107472732, −6.39907136659760764864025381384, −6.03140685239265504667261248223, −5.18790578242403276907541975149, −5.02822315820220277643281862145, −4.00486338420961002866546391450, −3.02585719999520149622429646144, −1.59248687664043637863577571798, −1.05869620729122783900593829843, 1.05869620729122783900593829843, 1.59248687664043637863577571798, 3.02585719999520149622429646144, 4.00486338420961002866546391450, 5.02822315820220277643281862145, 5.18790578242403276907541975149, 6.03140685239265504667261248223, 6.39907136659760764864025381384, 7.57156544957932290468107472732, 7.956025303016115457145403627699

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