Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 401 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 0.223·3-s + 4-s + 5-s − 0.223·6-s − 0.465·7-s − 8-s − 2.94·9-s − 10-s + 6.17·11-s + 0.223·12-s − 4.22·13-s + 0.465·14-s + 0.223·15-s + 16-s + 1.20·17-s + 2.94·18-s − 2.12·19-s + 20-s − 0.104·21-s − 6.17·22-s − 0.690·23-s − 0.223·24-s + 25-s + 4.22·26-s − 1.33·27-s − 0.465·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.129·3-s + 0.5·4-s + 0.447·5-s − 0.0913·6-s − 0.175·7-s − 0.353·8-s − 0.983·9-s − 0.316·10-s + 1.86·11-s + 0.0645·12-s − 1.17·13-s + 0.124·14-s + 0.0577·15-s + 0.250·16-s + 0.291·17-s + 0.695·18-s − 0.487·19-s + 0.223·20-s − 0.0227·21-s − 1.31·22-s − 0.143·23-s − 0.0456·24-s + 0.200·25-s + 0.827·26-s − 0.256·27-s − 0.0879·28-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 \( 1 - 0.223T + 3T^{2} \)
7 \( 1 + 0.465T + 7T^{2} \)
11 \( 1 - 6.17T + 11T^{2} \)
13 \( 1 + 4.22T + 13T^{2} \)
17 \( 1 - 1.20T + 17T^{2} \)
19 \( 1 + 2.12T + 19T^{2} \)
23 \( 1 + 0.690T + 23T^{2} \)
29 \( 1 + 9.84T + 29T^{2} \)
31 \( 1 - 1.38T + 31T^{2} \)
37 \( 1 - 6.67T + 37T^{2} \)
41 \( 1 + 7.06T + 41T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 - 2.50T + 47T^{2} \)
53 \( 1 + 3.84T + 53T^{2} \)
59 \( 1 + 9.21T + 59T^{2} \)
61 \( 1 + 6.81T + 61T^{2} \)
67 \( 1 - 5.63T + 67T^{2} \)
71 \( 1 + 3.71T + 71T^{2} \)
73 \( 1 - 2.89T + 73T^{2} \)
79 \( 1 + 0.949T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 + 7.25T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.134754261284575609672143221955, −7.45153543039591581389538559971, −6.57534705797438133818877608873, −6.10553643517870487866763564595, −5.24330349326851829115727929835, −4.15389455218078910239407305482, −3.24724735339100871088376678597, −2.31396593164301114945011574558, −1.42440661892480559643153495973, 0, 1.42440661892480559643153495973, 2.31396593164301114945011574558, 3.24724735339100871088376678597, 4.15389455218078910239407305482, 5.24330349326851829115727929835, 6.10553643517870487866763564595, 6.57534705797438133818877608873, 7.45153543039591581389538559971, 8.134754261284575609672143221955

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