Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.84·3-s + 4-s + 5-s + 1.84·6-s + 0.526·7-s − 8-s + 0.399·9-s − 10-s + 11-s − 1.84·12-s + 5.89·13-s − 0.526·14-s − 1.84·15-s + 16-s − 7.20·17-s − 0.399·18-s − 7.63·19-s + 20-s − 0.971·21-s − 22-s + 3.89·23-s + 1.84·24-s + 25-s − 5.89·26-s + 4.79·27-s + 0.526·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.06·3-s + 0.5·4-s + 0.447·5-s + 0.752·6-s + 0.199·7-s − 0.353·8-s + 0.133·9-s − 0.316·10-s + 0.301·11-s − 0.532·12-s + 1.63·13-s − 0.140·14-s − 0.476·15-s + 0.250·16-s − 1.74·17-s − 0.0941·18-s − 1.75·19-s + 0.223·20-s − 0.211·21-s − 0.213·22-s + 0.811·23-s + 0.376·24-s + 0.200·25-s − 1.15·26-s + 0.922·27-s + 0.0995·28-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  =  1
Selberg data  =  $(2,\ 4730,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 1.84T + 3T^{2} \)
7 \( 1 - 0.526T + 7T^{2} \)
13 \( 1 - 5.89T + 13T^{2} \)
17 \( 1 + 7.20T + 17T^{2} \)
19 \( 1 + 7.63T + 19T^{2} \)
23 \( 1 - 3.89T + 23T^{2} \)
29 \( 1 - 8.04T + 29T^{2} \)
31 \( 1 - 2.80T + 31T^{2} \)
37 \( 1 + 5.62T + 37T^{2} \)
41 \( 1 + 1.80T + 41T^{2} \)
47 \( 1 + 8.47T + 47T^{2} \)
53 \( 1 + 6.63T + 53T^{2} \)
59 \( 1 + 2.91T + 59T^{2} \)
61 \( 1 + 0.160T + 61T^{2} \)
67 \( 1 + 1.49T + 67T^{2} \)
71 \( 1 - 9.81T + 71T^{2} \)
73 \( 1 + 0.652T + 73T^{2} \)
79 \( 1 + 9.25T + 79T^{2} \)
83 \( 1 - 0.697T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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

−8.343609872225976709204589322319, −6.84340461282054555617412498413, −6.39154243612876742264281933505, −6.19270469744727683994071741274, −5.00099653665708971193448429773, −4.41595787356889347287177101047, −3.21604207676774996734985454210, −2.09417507691815021202547598700, −1.19161153101949589856171613130, 0, 1.19161153101949589856171613130, 2.09417507691815021202547598700, 3.21604207676774996734985454210, 4.41595787356889347287177101047, 5.00099653665708971193448429773, 6.19270469744727683994071741274, 6.39154243612876742264281933505, 6.84340461282054555617412498413, 8.343609872225976709204589322319

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