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 + 0.271·3-s + 4-s − 5-s − 0.271·6-s + 1.75·7-s − 8-s − 2.92·9-s + 10-s + 0.0291·11-s + 0.271·12-s + 5.25·13-s − 1.75·14-s − 0.271·15-s + 16-s − 3.06·17-s + 2.92·18-s − 6.99·19-s − 20-s + 0.477·21-s − 0.0291·22-s + 3.94·23-s − 0.271·24-s + 25-s − 5.25·26-s − 1.60·27-s + 1.75·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.156·3-s + 0.5·4-s − 0.447·5-s − 0.110·6-s + 0.665·7-s − 0.353·8-s − 0.975·9-s + 0.316·10-s + 0.00878·11-s + 0.0783·12-s + 1.45·13-s − 0.470·14-s − 0.0701·15-s + 0.250·16-s − 0.744·17-s + 0.689·18-s − 1.60·19-s − 0.223·20-s + 0.104·21-s − 0.00620·22-s + 0.823·23-s − 0.0554·24-s + 0.200·25-s − 1.03·26-s − 0.309·27-s + 0.332·28-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 \( 1 - 0.271T + 3T^{2} \)
7 \( 1 - 1.75T + 7T^{2} \)
11 \( 1 - 0.0291T + 11T^{2} \)
13 \( 1 - 5.25T + 13T^{2} \)
17 \( 1 + 3.06T + 17T^{2} \)
19 \( 1 + 6.99T + 19T^{2} \)
23 \( 1 - 3.94T + 23T^{2} \)
29 \( 1 - 8.01T + 29T^{2} \)
31 \( 1 + 7.61T + 31T^{2} \)
37 \( 1 + 3.48T + 37T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 - 3.83T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 - 1.28T + 59T^{2} \)
61 \( 1 + 5.25T + 61T^{2} \)
67 \( 1 + 2.24T + 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 + 2.43T + 79T^{2} \)
83 \( 1 - 8.86T + 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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

−8.332189134220904406137302653508, −7.56329516745542786610030002023, −6.60623707583315976647944706807, −6.11248685454691414789994068140, −5.09509773860663945414231139701, −4.18810977554811713739587018087, −3.30501885960915770936615965949, −2.35117619110972442694153945615, −1.34151703321977202334308445314, 0, 1.34151703321977202334308445314, 2.35117619110972442694153945615, 3.30501885960915770936615965949, 4.18810977554811713739587018087, 5.09509773860663945414231139701, 6.11248685454691414789994068140, 6.60623707583315976647944706807, 7.56329516745542786610030002023, 8.332189134220904406137302653508

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