Properties

Label 2-4026-1.1-c1-0-100
Degree $2$
Conductor $4026$
Sign $-1$
Analytic cond. $32.1477$
Root an. cond. $5.66990$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 1.14·5-s + 6-s − 1.54·7-s + 8-s + 9-s + 1.14·10-s − 11-s + 12-s − 6.01·13-s − 1.54·14-s + 1.14·15-s + 16-s − 5.63·17-s + 18-s − 2.94·19-s + 1.14·20-s − 1.54·21-s − 22-s − 3.47·23-s + 24-s − 3.69·25-s − 6.01·26-s + 27-s − 1.54·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.511·5-s + 0.408·6-s − 0.584·7-s + 0.353·8-s + 0.333·9-s + 0.361·10-s − 0.301·11-s + 0.288·12-s − 1.66·13-s − 0.413·14-s + 0.295·15-s + 0.250·16-s − 1.36·17-s + 0.235·18-s − 0.675·19-s + 0.255·20-s − 0.337·21-s − 0.213·22-s − 0.725·23-s + 0.204·24-s − 0.738·25-s − 1.18·26-s + 0.192·27-s − 0.292·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4026\)    =    \(2 \cdot 3 \cdot 11 \cdot 61\)
Sign: $-1$
Analytic conductor: \(32.1477\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4026,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 + T \)
61 \( 1 + T \)
good5 \( 1 - 1.14T + 5T^{2} \)
7 \( 1 + 1.54T + 7T^{2} \)
13 \( 1 + 6.01T + 13T^{2} \)
17 \( 1 + 5.63T + 17T^{2} \)
19 \( 1 + 2.94T + 19T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 + 2.10T + 29T^{2} \)
31 \( 1 + 4.77T + 31T^{2} \)
37 \( 1 - 6.21T + 37T^{2} \)
41 \( 1 + 1.67T + 41T^{2} \)
43 \( 1 - 5.01T + 43T^{2} \)
47 \( 1 + 4.26T + 47T^{2} \)
53 \( 1 - 2.04T + 53T^{2} \)
59 \( 1 + 6.35T + 59T^{2} \)
67 \( 1 - 1.66T + 67T^{2} \)
71 \( 1 - 8.23T + 71T^{2} \)
73 \( 1 - 3.82T + 73T^{2} \)
79 \( 1 - 2.20T + 79T^{2} \)
83 \( 1 + 4.41T + 83T^{2} \)
89 \( 1 - 1.84T + 89T^{2} \)
97 \( 1 - 8.66T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.897901321236244994841813858891, −7.31167440103198892209576944858, −6.51843832369090079520249354499, −5.90557040180825552084621218104, −4.92976604872709052147544154572, −4.31013703724824540958673281826, −3.39921135594231060865697817838, −2.34506872001969458374719879997, −2.07374107656679427706389619976, 0, 2.07374107656679427706389619976, 2.34506872001969458374719879997, 3.39921135594231060865697817838, 4.31013703724824540958673281826, 4.92976604872709052147544154572, 5.90557040180825552084621218104, 6.51843832369090079520249354499, 7.31167440103198892209576944858, 7.897901321236244994841813858891

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