Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 401 $
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 − 2.86·3-s + 4-s − 5-s + 2.86·6-s + 0.981·7-s − 8-s + 5.21·9-s + 10-s + 3.23·11-s − 2.86·12-s − 5.40·13-s − 0.981·14-s + 2.86·15-s + 16-s − 3.01·17-s − 5.21·18-s + 4.93·19-s − 20-s − 2.81·21-s − 3.23·22-s + 1.30·23-s + 2.86·24-s + 25-s + 5.40·26-s − 6.34·27-s + 0.981·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.65·3-s + 0.5·4-s − 0.447·5-s + 1.17·6-s + 0.370·7-s − 0.353·8-s + 1.73·9-s + 0.316·10-s + 0.975·11-s − 0.827·12-s − 1.49·13-s − 0.262·14-s + 0.740·15-s + 0.250·16-s − 0.731·17-s − 1.22·18-s + 1.13·19-s − 0.223·20-s − 0.613·21-s − 0.690·22-s + 0.271·23-s + 0.585·24-s + 0.200·25-s + 1.05·26-s − 1.22·27-s + 0.185·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4010\)    =    \(2 \cdot 5 \cdot 401\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4010} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4010,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;401\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;401\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
5 \( 1 + T \)
401 \( 1 + T \)
good3 \( 1 + 2.86T + 3T^{2} \)
7 \( 1 - 0.981T + 7T^{2} \)
11 \( 1 - 3.23T + 11T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
17 \( 1 + 3.01T + 17T^{2} \)
19 \( 1 - 4.93T + 19T^{2} \)
23 \( 1 - 1.30T + 23T^{2} \)
29 \( 1 + 4.83T + 29T^{2} \)
31 \( 1 + 1.22T + 31T^{2} \)
37 \( 1 + 1.55T + 37T^{2} \)
41 \( 1 - 3.26T + 41T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 + 3.38T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 - 15.0T + 61T^{2} \)
67 \( 1 + 4.93T + 67T^{2} \)
71 \( 1 + 0.116T + 71T^{2} \)
73 \( 1 - 4.74T + 73T^{2} \)
79 \( 1 + 0.914T + 79T^{2} \)
83 \( 1 - 7.48T + 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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

−7.84566114631201486642038122048, −7.24906478306315004966517926720, −6.74872388659980368164276902774, −5.93861543616459008116505094135, −5.09625153256857087509830483257, −4.58675728659990194922583953676, −3.49814494130326947372406443804, −2.09177702485041161397186369010, −1.01811156127688130843557216962, 0, 1.01811156127688130843557216962, 2.09177702485041161397186369010, 3.49814494130326947372406443804, 4.58675728659990194922583953676, 5.09625153256857087509830483257, 5.93861543616459008116505094135, 6.74872388659980368164276902774, 7.24906478306315004966517926720, 7.84566114631201486642038122048

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