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 − 1.74·3-s + 4-s + 5-s + 1.74·6-s + 2.79·7-s − 8-s + 0.0402·9-s − 10-s + 2.90·11-s − 1.74·12-s + 3.29·13-s − 2.79·14-s − 1.74·15-s + 16-s − 4.26·17-s − 0.0402·18-s − 0.133·19-s + 20-s − 4.88·21-s − 2.90·22-s − 1.71·23-s + 1.74·24-s + 25-s − 3.29·26-s + 5.16·27-s + 2.79·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.00·3-s + 0.5·4-s + 0.447·5-s + 0.711·6-s + 1.05·7-s − 0.353·8-s + 0.0134·9-s − 0.316·10-s + 0.876·11-s − 0.503·12-s + 0.914·13-s − 0.748·14-s − 0.450·15-s + 0.250·16-s − 1.03·17-s − 0.00948·18-s − 0.0305·19-s + 0.223·20-s − 1.06·21-s − 0.619·22-s − 0.356·23-s + 0.355·24-s + 0.200·25-s − 0.646·26-s + 0.993·27-s + 0.528·28-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 \( 1 + 1.74T + 3T^{2} \)
7 \( 1 - 2.79T + 7T^{2} \)
11 \( 1 - 2.90T + 11T^{2} \)
13 \( 1 - 3.29T + 13T^{2} \)
17 \( 1 + 4.26T + 17T^{2} \)
19 \( 1 + 0.133T + 19T^{2} \)
23 \( 1 + 1.71T + 23T^{2} \)
29 \( 1 + 2.64T + 29T^{2} \)
31 \( 1 + 9.13T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 + 5.26T + 41T^{2} \)
43 \( 1 + 4.56T + 43T^{2} \)
47 \( 1 - 6.49T + 47T^{2} \)
53 \( 1 - 4.03T + 53T^{2} \)
59 \( 1 + 6.96T + 59T^{2} \)
61 \( 1 + 5.75T + 61T^{2} \)
67 \( 1 + 1.15T + 67T^{2} \)
71 \( 1 + 0.443T + 71T^{2} \)
73 \( 1 + 0.955T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 8.57T + 97T^{2} \)
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

−8.310931264852036419769249865584, −7.19544657993500368293944926705, −6.65965964136783610259906346935, −5.87103839757841456013532536133, −5.33146196045535496981857754241, −4.41206555336677188955922393837, −3.41823327839763969706561801379, −1.95860501700622797059490697609, −1.39853409528883700262324910648, 0, 1.39853409528883700262324910648, 1.95860501700622797059490697609, 3.41823327839763969706561801379, 4.41206555336677188955922393837, 5.33146196045535496981857754241, 5.87103839757841456013532536133, 6.65965964136783610259906346935, 7.19544657993500368293944926705, 8.310931264852036419769249865584

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