Properties

Label 2-5054-1.1-c1-0-110
Degree $2$
Conductor $5054$
Sign $-1$
Analytic cond. $40.3563$
Root an. cond. $6.35266$
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 − 2.17·3-s + 4-s + 3.52·5-s + 2.17·6-s + 7-s − 8-s + 1.73·9-s − 3.52·10-s − 2.80·11-s − 2.17·12-s − 1.34·13-s − 14-s − 7.65·15-s + 16-s − 3.69·17-s − 1.73·18-s + 3.52·20-s − 2.17·21-s + 2.80·22-s + 4.80·23-s + 2.17·24-s + 7.39·25-s + 1.34·26-s + 2.75·27-s + 28-s − 5.35·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.25·3-s + 0.5·4-s + 1.57·5-s + 0.888·6-s + 0.377·7-s − 0.353·8-s + 0.577·9-s − 1.11·10-s − 0.845·11-s − 0.628·12-s − 0.372·13-s − 0.267·14-s − 1.97·15-s + 0.250·16-s − 0.896·17-s − 0.408·18-s + 0.787·20-s − 0.474·21-s + 0.597·22-s + 1.00·23-s + 0.444·24-s + 1.47·25-s + 0.263·26-s + 0.530·27-s + 0.188·28-s − 0.993·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5054\)    =    \(2 \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(40.3563\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5054} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5054,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 - T \)
19 \( 1 \)
good3 \( 1 + 2.17T + 3T^{2} \)
5 \( 1 - 3.52T + 5T^{2} \)
11 \( 1 + 2.80T + 11T^{2} \)
13 \( 1 + 1.34T + 13T^{2} \)
17 \( 1 + 3.69T + 17T^{2} \)
23 \( 1 - 4.80T + 23T^{2} \)
29 \( 1 + 5.35T + 29T^{2} \)
31 \( 1 - 3.32T + 31T^{2} \)
37 \( 1 + 9.91T + 37T^{2} \)
41 \( 1 - 6.54T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + 9.44T + 47T^{2} \)
53 \( 1 - 2.46T + 53T^{2} \)
59 \( 1 + 0.501T + 59T^{2} \)
61 \( 1 + 9.22T + 61T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 1.92T + 73T^{2} \)
79 \( 1 + 17.5T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + 16.5T + 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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

−7.79258942278240009907932084067, −7.01394204209777444762218216420, −6.39013796786194019764870263737, −5.69996157972434381085509808231, −5.25220196868334774423815526037, −4.53857457425808245355643015855, −2.90173028449849663188531584096, −2.14521828909898048430111621300, −1.24288302386602278242716964416, 0, 1.24288302386602278242716964416, 2.14521828909898048430111621300, 2.90173028449849663188531584096, 4.53857457425808245355643015855, 5.25220196868334774423815526037, 5.69996157972434381085509808231, 6.39013796786194019764870263737, 7.01394204209777444762218216420, 7.79258942278240009907932084067

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