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 + 3.42·3-s + 4-s + 5-s − 3.42·6-s − 0.184·7-s − 8-s + 8.75·9-s − 10-s + 11-s + 3.42·12-s − 1.86·13-s + 0.184·14-s + 3.42·15-s + 16-s + 1.38·17-s − 8.75·18-s + 2.95·19-s + 20-s − 0.632·21-s − 22-s + 4.77·23-s − 3.42·24-s + 25-s + 1.86·26-s + 19.7·27-s − 0.184·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.97·3-s + 0.5·4-s + 0.447·5-s − 1.39·6-s − 0.0697·7-s − 0.353·8-s + 2.91·9-s − 0.316·10-s + 0.301·11-s + 0.989·12-s − 0.517·13-s + 0.0493·14-s + 0.885·15-s + 0.250·16-s + 0.335·17-s − 2.06·18-s + 0.677·19-s + 0.223·20-s − 0.138·21-s − 0.213·22-s + 0.996·23-s − 0.699·24-s + 0.200·25-s + 0.366·26-s + 3.79·27-s − 0.0348·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.674542925$
$L(\frac12)$  $\approx$  $3.674542925$
$L(\frac{3}{2})$   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 \( 1 - 3.42T + 3T^{2} \)
7 \( 1 + 0.184T + 7T^{2} \)
13 \( 1 + 1.86T + 13T^{2} \)
17 \( 1 - 1.38T + 17T^{2} \)
19 \( 1 - 2.95T + 19T^{2} \)
23 \( 1 - 4.77T + 23T^{2} \)
29 \( 1 + 2.99T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 + 8.37T + 37T^{2} \)
41 \( 1 + 4.20T + 41T^{2} \)
47 \( 1 + 8.11T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 - 0.668T + 59T^{2} \)
61 \( 1 + 1.18T + 61T^{2} \)
67 \( 1 + 7.72T + 67T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + 17.0T + 79T^{2} \)
83 \( 1 - 9.61T + 83T^{2} \)
89 \( 1 - 9.80T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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

−8.328990593885953530039722723615, −7.83343428763882529186371877556, −7.03566917274230610897449591236, −6.62487934862621037005992031948, −5.26733877295042773249929999349, −4.38381640002133078442721804037, −3.29489864256822046371568388319, −2.89608203071631052191299811747, −1.93731653990956197826984733401, −1.18589860329243067506047997612, 1.18589860329243067506047997612, 1.93731653990956197826984733401, 2.89608203071631052191299811747, 3.29489864256822046371568388319, 4.38381640002133078442721804037, 5.26733877295042773249929999349, 6.62487934862621037005992031948, 7.03566917274230610897449591236, 7.83343428763882529186371877556, 8.328990593885953530039722723615

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