Properties

Label 2-4030-1.1-c1-0-61
Degree $2$
Conductor $4030$
Sign $-1$
Analytic cond. $32.1797$
Root an. cond. $5.67271$
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 − 2.73·3-s + 4-s + 5-s + 2.73·6-s − 0.880·7-s − 8-s + 4.49·9-s − 10-s + 2.88·11-s − 2.73·12-s + 13-s + 0.880·14-s − 2.73·15-s + 16-s − 2.60·17-s − 4.49·18-s − 3.20·19-s + 20-s + 2.41·21-s − 2.88·22-s + 2.33·23-s + 2.73·24-s + 25-s − 26-s − 4.10·27-s − 0.880·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.58·3-s + 0.5·4-s + 0.447·5-s + 1.11·6-s − 0.332·7-s − 0.353·8-s + 1.49·9-s − 0.316·10-s + 0.870·11-s − 0.790·12-s + 0.277·13-s + 0.235·14-s − 0.707·15-s + 0.250·16-s − 0.631·17-s − 1.06·18-s − 0.736·19-s + 0.223·20-s + 0.525·21-s − 0.615·22-s + 0.487·23-s + 0.558·24-s + 0.200·25-s − 0.196·26-s − 0.789·27-s − 0.166·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
Sign: $-1$
Analytic conductor: \(32.1797\)
Root analytic conductor: \(5.67271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4030,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 - T \)
31 \( 1 + T \)
good3 \( 1 + 2.73T + 3T^{2} \)
7 \( 1 + 0.880T + 7T^{2} \)
11 \( 1 - 2.88T + 11T^{2} \)
17 \( 1 + 2.60T + 17T^{2} \)
19 \( 1 + 3.20T + 19T^{2} \)
23 \( 1 - 2.33T + 23T^{2} \)
29 \( 1 + 0.826T + 29T^{2} \)
37 \( 1 + 9.57T + 37T^{2} \)
41 \( 1 - 6.05T + 41T^{2} \)
43 \( 1 - 8.96T + 43T^{2} \)
47 \( 1 + 8.62T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 6.57T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 2.61T + 67T^{2} \)
71 \( 1 - 6.23T + 71T^{2} \)
73 \( 1 + 4.70T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 8.09T + 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + 4.56T + 97T^{2} \)
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

−8.078738018472922343897470913750, −7.06637789757142950118120778450, −6.45874065139813188900840060005, −6.17343263612869962342177277705, −5.24974406640901775677550972814, −4.48111811744317232394597228678, −3.43619936265758172393224459994, −2.06677484420629230808926127120, −1.12380967815255804719658173558, 0, 1.12380967815255804719658173558, 2.06677484420629230808926127120, 3.43619936265758172393224459994, 4.48111811744317232394597228678, 5.24974406640901775677550972814, 6.17343263612869962342177277705, 6.45874065139813188900840060005, 7.06637789757142950118120778450, 8.078738018472922343897470913750

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