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 + 1.70·3-s + 4-s + 5-s − 1.70·6-s + 0.961·7-s − 8-s − 0.103·9-s − 10-s + 11-s + 1.70·12-s − 2.37·13-s − 0.961·14-s + 1.70·15-s + 16-s + 0.206·17-s + 0.103·18-s − 0.413·19-s + 20-s + 1.63·21-s − 22-s − 4.37·23-s − 1.70·24-s + 25-s + 2.37·26-s − 5.28·27-s + 0.961·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.982·3-s + 0.5·4-s + 0.447·5-s − 0.694·6-s + 0.363·7-s − 0.353·8-s − 0.0343·9-s − 0.316·10-s + 0.301·11-s + 0.491·12-s − 0.658·13-s − 0.257·14-s + 0.439·15-s + 0.250·16-s + 0.0500·17-s + 0.0243·18-s − 0.0949·19-s + 0.223·20-s + 0.357·21-s − 0.213·22-s − 0.912·23-s − 0.347·24-s + 0.200·25-s + 0.465·26-s − 1.01·27-s + 0.181·28-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 \( 1 - 1.70T + 3T^{2} \)
7 \( 1 - 0.961T + 7T^{2} \)
13 \( 1 + 2.37T + 13T^{2} \)
17 \( 1 - 0.206T + 17T^{2} \)
19 \( 1 + 0.413T + 19T^{2} \)
23 \( 1 + 4.37T + 23T^{2} \)
29 \( 1 + 5.05T + 29T^{2} \)
31 \( 1 + 6.59T + 31T^{2} \)
37 \( 1 + 9.05T + 37T^{2} \)
41 \( 1 + 7.15T + 41T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 - 9.81T + 53T^{2} \)
59 \( 1 - 5.21T + 59T^{2} \)
61 \( 1 - 7.54T + 61T^{2} \)
67 \( 1 - 7.11T + 67T^{2} \)
71 \( 1 - 1.63T + 71T^{2} \)
73 \( 1 + 9.11T + 73T^{2} \)
79 \( 1 - 2.19T + 79T^{2} \)
83 \( 1 + 5.54T + 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 - 2.79T + 97T^{2} \)
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

−8.184203812720445212890677756934, −7.33787035814286041444212430234, −6.78903572284254597975526226664, −5.74641360790131176535381365120, −5.12104481349279086824938424779, −3.88496911731922832737247520151, −3.19966182872137490508014728116, −2.13535643350076355741482496870, −1.71580414946402963754479911383, 0, 1.71580414946402963754479911383, 2.13535643350076355741482496870, 3.19966182872137490508014728116, 3.88496911731922832737247520151, 5.12104481349279086824938424779, 5.74641360790131176535381365120, 6.78903572284254597975526226664, 7.33787035814286041444212430234, 8.184203812720445212890677756934

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