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 + 1.57·3-s + 4-s + 5-s − 1.57·6-s + 0.800·7-s − 8-s − 0.514·9-s − 10-s + 11-s + 1.57·12-s − 1.02·13-s − 0.800·14-s + 1.57·15-s + 16-s − 5.32·17-s + 0.514·18-s − 2.64·19-s + 20-s + 1.26·21-s − 22-s − 3.02·23-s − 1.57·24-s + 25-s + 1.02·26-s − 5.54·27-s + 0.800·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.910·3-s + 0.5·4-s + 0.447·5-s − 0.643·6-s + 0.302·7-s − 0.353·8-s − 0.171·9-s − 0.316·10-s + 0.301·11-s + 0.455·12-s − 0.284·13-s − 0.213·14-s + 0.407·15-s + 0.250·16-s − 1.29·17-s + 0.121·18-s − 0.607·19-s + 0.223·20-s + 0.275·21-s − 0.213·22-s − 0.630·23-s − 0.321·24-s + 0.200·25-s + 0.201·26-s − 1.06·27-s + 0.151·28-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 \( 1 - 1.57T + 3T^{2} \)
7 \( 1 - 0.800T + 7T^{2} \)
13 \( 1 + 1.02T + 13T^{2} \)
17 \( 1 + 5.32T + 17T^{2} \)
19 \( 1 + 2.64T + 19T^{2} \)
23 \( 1 + 3.02T + 23T^{2} \)
29 \( 1 + 0.208T + 29T^{2} \)
31 \( 1 - 1.97T + 31T^{2} \)
37 \( 1 + 6.73T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
47 \( 1 - 3.55T + 47T^{2} \)
53 \( 1 + 9.61T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 + 2.97T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + 6.73T + 71T^{2} \)
73 \( 1 - 1.32T + 73T^{2} \)
79 \( 1 - 8.33T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 - 9.34T + 89T^{2} \)
97 \( 1 - 0.294T + 97T^{2} \)
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

−8.018402176100685791770487947160, −7.50727716793045095138377687787, −6.50333363356265948199766734985, −6.04300080141468893816870464479, −4.91007392317356628783647988498, −4.07392764730143804197945854617, −3.03836145619920466371699854399, −2.28302236346254022619138946966, −1.61751233852517058776895457527, 0, 1.61751233852517058776895457527, 2.28302236346254022619138946966, 3.03836145619920466371699854399, 4.07392764730143804197945854617, 4.91007392317356628783647988498, 6.04300080141468893816870464479, 6.50333363356265948199766734985, 7.50727716793045095138377687787, 8.018402176100685791770487947160

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