Properties

Label 2-5070-1.1-c1-0-31
Degree $2$
Conductor $5070$
Sign $1$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
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 − 3-s + 4-s + 5-s + 6-s + 2.55·7-s − 8-s + 9-s − 10-s + 2.24·11-s − 12-s − 2.55·14-s − 15-s + 16-s + 2.02·17-s − 18-s − 1.33·19-s + 20-s − 2.55·21-s − 2.24·22-s + 5.58·23-s + 24-s + 25-s − 27-s + 2.55·28-s + 3.26·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.965·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.677·11-s − 0.288·12-s − 0.682·14-s − 0.258·15-s + 0.250·16-s + 0.490·17-s − 0.235·18-s − 0.306·19-s + 0.223·20-s − 0.557·21-s − 0.479·22-s + 1.16·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.482·28-s + 0.606·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.628592392\)
\(L(\frac12)\) \(\approx\) \(1.628592392\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 - 2.55T + 7T^{2} \)
11 \( 1 - 2.24T + 11T^{2} \)
17 \( 1 - 2.02T + 17T^{2} \)
19 \( 1 + 1.33T + 19T^{2} \)
23 \( 1 - 5.58T + 23T^{2} \)
29 \( 1 - 3.26T + 29T^{2} \)
31 \( 1 + 1.35T + 31T^{2} \)
37 \( 1 - 9.64T + 37T^{2} \)
41 \( 1 - 1.06T + 41T^{2} \)
43 \( 1 - 0.137T + 43T^{2} \)
47 \( 1 + 12.7T + 47T^{2} \)
53 \( 1 + 3.03T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 + 0.972T + 61T^{2} \)
67 \( 1 - 8.51T + 67T^{2} \)
71 \( 1 - 2.69T + 71T^{2} \)
73 \( 1 - 1.48T + 73T^{2} \)
79 \( 1 + 7.77T + 79T^{2} \)
83 \( 1 - 17.7T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 7.14T + 97T^{2} \)
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

−8.189927110023359231292118961198, −7.64143007298733193025077096260, −6.69419979985273590605932872251, −6.29582689258910157218267742706, −5.29081711185365225161889679601, −4.77747145864549295479881626556, −3.73125113921214843641479629738, −2.59426246880366383518821996085, −1.59279655512156638292963990338, −0.871005823326946464219427606020, 0.871005823326946464219427606020, 1.59279655512156638292963990338, 2.59426246880366383518821996085, 3.73125113921214843641479629738, 4.77747145864549295479881626556, 5.29081711185365225161889679601, 6.29582689258910157218267742706, 6.69419979985273590605932872251, 7.64143007298733193025077096260, 8.189927110023359231292118961198

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