Properties

Label 2-6010-1.1-c1-0-110
Degree $2$
Conductor $6010$
Sign $-1$
Analytic cond. $47.9900$
Root an. cond. $6.92748$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.98·3-s + 4-s − 5-s − 1.98·6-s − 1.31·7-s + 8-s + 0.924·9-s − 10-s − 3.29·11-s − 1.98·12-s − 2.77·13-s − 1.31·14-s + 1.98·15-s + 16-s + 5.53·17-s + 0.924·18-s + 8.26·19-s − 20-s + 2.60·21-s − 3.29·22-s − 3.94·23-s − 1.98·24-s + 25-s − 2.77·26-s + 4.11·27-s − 1.31·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.14·3-s + 0.5·4-s − 0.447·5-s − 0.808·6-s − 0.496·7-s + 0.353·8-s + 0.308·9-s − 0.316·10-s − 0.994·11-s − 0.571·12-s − 0.770·13-s − 0.351·14-s + 0.511·15-s + 0.250·16-s + 1.34·17-s + 0.217·18-s + 1.89·19-s − 0.223·20-s + 0.567·21-s − 0.703·22-s − 0.821·23-s − 0.404·24-s + 0.200·25-s − 0.544·26-s + 0.791·27-s − 0.248·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6010\)    =    \(2 \cdot 5 \cdot 601\)
Sign: $-1$
Analytic conductor: \(47.9900\)
Root analytic conductor: \(6.92748\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6010,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
601 \( 1 - T \)
good3 \( 1 + 1.98T + 3T^{2} \)
7 \( 1 + 1.31T + 7T^{2} \)
11 \( 1 + 3.29T + 11T^{2} \)
13 \( 1 + 2.77T + 13T^{2} \)
17 \( 1 - 5.53T + 17T^{2} \)
19 \( 1 - 8.26T + 19T^{2} \)
23 \( 1 + 3.94T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 - 8.12T + 31T^{2} \)
37 \( 1 - 3.13T + 37T^{2} \)
41 \( 1 - 6.16T + 41T^{2} \)
43 \( 1 - 7.50T + 43T^{2} \)
47 \( 1 + 2.19T + 47T^{2} \)
53 \( 1 - 1.42T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 - 7.06T + 61T^{2} \)
67 \( 1 + 7.92T + 67T^{2} \)
71 \( 1 - 1.57T + 71T^{2} \)
73 \( 1 - 7.48T + 73T^{2} \)
79 \( 1 + 8.41T + 79T^{2} \)
83 \( 1 + 1.48T + 83T^{2} \)
89 \( 1 + 3.59T + 89T^{2} \)
97 \( 1 + 4.31T + 97T^{2} \)
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

−7.73135272687699467059274979995, −6.94000023769205070258900646720, −5.97328113551352570519212978439, −5.57117628340429889744850119353, −5.05141300783274971985634576760, −4.19677076915469585698194412668, −3.22519328273998456007970136349, −2.64525759496355147520733593151, −1.13064560815708644289777482735, 0, 1.13064560815708644289777482735, 2.64525759496355147520733593151, 3.22519328273998456007970136349, 4.19677076915469585698194412668, 5.05141300783274971985634576760, 5.57117628340429889744850119353, 5.97328113551352570519212978439, 6.94000023769205070258900646720, 7.73135272687699467059274979995

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