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 + 0.675·3-s + 4-s + 5-s − 0.675·6-s − 1.96·7-s − 8-s − 2.54·9-s − 10-s − 0.702·11-s + 0.675·12-s + 3.63·13-s + 1.96·14-s + 0.675·15-s + 16-s − 0.463·17-s + 2.54·18-s + 1.64·19-s + 20-s − 1.32·21-s + 0.702·22-s − 3.53·23-s − 0.675·24-s + 25-s − 3.63·26-s − 3.74·27-s − 1.96·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.390·3-s + 0.5·4-s + 0.447·5-s − 0.275·6-s − 0.743·7-s − 0.353·8-s − 0.847·9-s − 0.316·10-s − 0.211·11-s + 0.195·12-s + 1.00·13-s + 0.525·14-s + 0.174·15-s + 0.250·16-s − 0.112·17-s + 0.599·18-s + 0.376·19-s + 0.223·20-s − 0.289·21-s + 0.149·22-s − 0.736·23-s − 0.137·24-s + 0.200·25-s − 0.713·26-s − 0.720·27-s − 0.371·28-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
401 \( 1 - T \)
good3 \( 1 - 0.675T + 3T^{2} \)
7 \( 1 + 1.96T + 7T^{2} \)
11 \( 1 + 0.702T + 11T^{2} \)
13 \( 1 - 3.63T + 13T^{2} \)
17 \( 1 + 0.463T + 17T^{2} \)
19 \( 1 - 1.64T + 19T^{2} \)
23 \( 1 + 3.53T + 23T^{2} \)
29 \( 1 - 2.53T + 29T^{2} \)
31 \( 1 + 6.10T + 31T^{2} \)
37 \( 1 - 5.76T + 37T^{2} \)
41 \( 1 - 7.14T + 41T^{2} \)
43 \( 1 + 6.31T + 43T^{2} \)
47 \( 1 - 5.97T + 47T^{2} \)
53 \( 1 + 4.57T + 53T^{2} \)
59 \( 1 + 2.56T + 59T^{2} \)
61 \( 1 + 9.20T + 61T^{2} \)
67 \( 1 - 2.71T + 67T^{2} \)
71 \( 1 + 3.57T + 71T^{2} \)
73 \( 1 + 1.05T + 73T^{2} \)
79 \( 1 + 7.77T + 79T^{2} \)
83 \( 1 + 5.64T + 83T^{2} \)
89 \( 1 - 2.42T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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

−8.122318598879485376304102134084, −7.59788256847260969921965391570, −6.51725469135678926146204039111, −6.06554697067360742942212546251, −5.35486077454489779793914742262, −4.02869067176157377170357205750, −3.16206429490263436149127001606, −2.49561102606008902058867743273, −1.38714389737662786671511790870, 0, 1.38714389737662786671511790870, 2.49561102606008902058867743273, 3.16206429490263436149127001606, 4.02869067176157377170357205750, 5.35486077454489779793914742262, 6.06554697067360742942212546251, 6.51725469135678926146204039111, 7.59788256847260969921965391570, 8.122318598879485376304102134084

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