Properties

Label 2-5054-1.1-c1-0-14
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.40·3-s + 4-s − 2.26·5-s + 1.40·6-s + 7-s − 8-s − 1.01·9-s + 2.26·10-s + 2.53·11-s − 1.40·12-s + 1.31·13-s − 14-s + 3.18·15-s + 16-s − 4.50·17-s + 1.01·18-s − 2.26·20-s − 1.40·21-s − 2.53·22-s − 7.70·23-s + 1.40·24-s + 0.126·25-s − 1.31·26-s + 5.65·27-s + 28-s + 1.79·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.812·3-s + 0.5·4-s − 1.01·5-s + 0.574·6-s + 0.377·7-s − 0.353·8-s − 0.339·9-s + 0.716·10-s + 0.764·11-s − 0.406·12-s + 0.364·13-s − 0.267·14-s + 0.822·15-s + 0.250·16-s − 1.09·17-s + 0.240·18-s − 0.506·20-s − 0.307·21-s − 0.540·22-s − 1.60·23-s + 0.287·24-s + 0.0253·25-s − 0.257·26-s + 1.08·27-s + 0.188·28-s + 0.333·29-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: \(0\)
Selberg data: \((2,\ 5054,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4721458800\)
\(L(\frac12)\) \(\approx\) \(0.4721458800\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.40T + 3T^{2} \)
5 \( 1 + 2.26T + 5T^{2} \)
11 \( 1 - 2.53T + 11T^{2} \)
13 \( 1 - 1.31T + 13T^{2} \)
17 \( 1 + 4.50T + 17T^{2} \)
23 \( 1 + 7.70T + 23T^{2} \)
29 \( 1 - 1.79T + 29T^{2} \)
31 \( 1 - 2.25T + 31T^{2} \)
37 \( 1 - 2.02T + 37T^{2} \)
41 \( 1 + 8.23T + 41T^{2} \)
43 \( 1 - 1.11T + 43T^{2} \)
47 \( 1 - 5.77T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 + 5.02T + 59T^{2} \)
61 \( 1 + 9.44T + 61T^{2} \)
67 \( 1 - 1.10T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 2.75T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 - 0.637T + 97T^{2} \)
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

−8.411862529802088005366569994961, −7.57718734061376781295202448112, −6.80242741531031725114306616227, −6.20971358027100581556526730555, −5.49613323671944383353740783580, −4.39993977671286668031661948039, −3.91900749991879004511808833618, −2.74398112880199557346757952366, −1.61684260431507643574646817898, −0.43834136206667458660224708644, 0.43834136206667458660224708644, 1.61684260431507643574646817898, 2.74398112880199557346757952366, 3.91900749991879004511808833618, 4.39993977671286668031661948039, 5.49613323671944383353740783580, 6.20971358027100581556526730555, 6.80242741531031725114306616227, 7.57718734061376781295202448112, 8.411862529802088005366569994961

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