Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 223 $
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 + 4-s + 3.02·5-s − 3.77·7-s + 8-s + 3.02·10-s − 1.20·11-s − 1.04·13-s − 3.77·14-s + 16-s − 3.48·17-s − 4.38·19-s + 3.02·20-s − 1.20·22-s − 6.93·23-s + 4.12·25-s − 1.04·26-s − 3.77·28-s + 1.44·29-s − 3.28·31-s + 32-s − 3.48·34-s − 11.3·35-s − 5.93·37-s − 4.38·38-s + 3.02·40-s − 5.58·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.35·5-s − 1.42·7-s + 0.353·8-s + 0.955·10-s − 0.362·11-s − 0.290·13-s − 1.00·14-s + 0.250·16-s − 0.844·17-s − 1.00·19-s + 0.675·20-s − 0.256·22-s − 1.44·23-s + 0.824·25-s − 0.205·26-s − 0.712·28-s + 0.268·29-s − 0.590·31-s + 0.176·32-s − 0.597·34-s − 1.92·35-s − 0.975·37-s − 0.712·38-s + 0.477·40-s − 0.872·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4014\)    =    \(2 \cdot 3^{2} \cdot 223\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4014} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4014,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;223\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;223\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
223 \( 1 + T \)
good5 \( 1 - 3.02T + 5T^{2} \)
7 \( 1 + 3.77T + 7T^{2} \)
11 \( 1 + 1.20T + 11T^{2} \)
13 \( 1 + 1.04T + 13T^{2} \)
17 \( 1 + 3.48T + 17T^{2} \)
19 \( 1 + 4.38T + 19T^{2} \)
23 \( 1 + 6.93T + 23T^{2} \)
29 \( 1 - 1.44T + 29T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 + 5.93T + 37T^{2} \)
41 \( 1 + 5.58T + 41T^{2} \)
43 \( 1 - 9.61T + 43T^{2} \)
47 \( 1 + 1.67T + 47T^{2} \)
53 \( 1 + 7.89T + 53T^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 - 7.69T + 67T^{2} \)
71 \( 1 + 2.66T + 71T^{2} \)
73 \( 1 - 3.48T + 73T^{2} \)
79 \( 1 + 7.37T + 79T^{2} \)
83 \( 1 - 2.11T + 83T^{2} \)
89 \( 1 - 6.47T + 89T^{2} \)
97 \( 1 - 8.21T + 97T^{2} \)
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

−8.027900632040529443544022340819, −6.97850066772823858231853284767, −6.37435295377596256897870516292, −5.99681506848008122852417763663, −5.20215737038215989748935838975, −4.26765570798430688114115023466, −3.37955323140548571450300994581, −2.45222773144984568089470746235, −1.88009574471048767165434544330, 0, 1.88009574471048767165434544330, 2.45222773144984568089470746235, 3.37955323140548571450300994581, 4.26765570798430688114115023466, 5.20215737038215989748935838975, 5.99681506848008122852417763663, 6.37435295377596256897870516292, 6.97850066772823858231853284767, 8.027900632040529443544022340819

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