Properties

Label 2-6042-1.1-c1-0-110
Degree $2$
Conductor $6042$
Sign $-1$
Analytic cond. $48.2456$
Root an. cond. $6.94590$
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 − 3-s + 4-s + 1.11·5-s + 6-s + 0.414·7-s − 8-s + 9-s − 1.11·10-s + 1.07·11-s − 12-s + 4.69·13-s − 0.414·14-s − 1.11·15-s + 16-s − 2.07·17-s − 18-s − 19-s + 1.11·20-s − 0.414·21-s − 1.07·22-s + 0.442·23-s + 24-s − 3.74·25-s − 4.69·26-s − 27-s + 0.414·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.500·5-s + 0.408·6-s + 0.156·7-s − 0.353·8-s + 0.333·9-s − 0.354·10-s + 0.323·11-s − 0.288·12-s + 1.30·13-s − 0.110·14-s − 0.289·15-s + 0.250·16-s − 0.504·17-s − 0.235·18-s − 0.229·19-s + 0.250·20-s − 0.0904·21-s − 0.229·22-s + 0.0922·23-s + 0.204·24-s − 0.749·25-s − 0.921·26-s − 0.192·27-s + 0.0783·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6042\)    =    \(2 \cdot 3 \cdot 19 \cdot 53\)
Sign: $-1$
Analytic conductor: \(48.2456\)
Root analytic conductor: \(6.94590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6042,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 + T \)
19 \( 1 + T \)
53 \( 1 + T \)
good5 \( 1 - 1.11T + 5T^{2} \)
7 \( 1 - 0.414T + 7T^{2} \)
11 \( 1 - 1.07T + 11T^{2} \)
13 \( 1 - 4.69T + 13T^{2} \)
17 \( 1 + 2.07T + 17T^{2} \)
23 \( 1 - 0.442T + 23T^{2} \)
29 \( 1 + 5.42T + 29T^{2} \)
31 \( 1 - 6.58T + 31T^{2} \)
37 \( 1 + 5.30T + 37T^{2} \)
41 \( 1 + 5.21T + 41T^{2} \)
43 \( 1 + 6.51T + 43T^{2} \)
47 \( 1 + 4.83T + 47T^{2} \)
59 \( 1 - 1.71T + 59T^{2} \)
61 \( 1 - 1.28T + 61T^{2} \)
67 \( 1 + 0.633T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + 0.429T + 73T^{2} \)
79 \( 1 + 4.95T + 79T^{2} \)
83 \( 1 - 1.90T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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

−7.81709381647979071707442340991, −6.87487728235781384856540194131, −6.37528518455242515637822985918, −5.79975334451199473487563313551, −4.96951381945429233890139956313, −4.02156848034438450402514522093, −3.17681756196313774934270219479, −1.91986902625605130382970634790, −1.34574726537964986671072293595, 0, 1.34574726537964986671072293595, 1.91986902625605130382970634790, 3.17681756196313774934270219479, 4.02156848034438450402514522093, 4.96951381945429233890139956313, 5.79975334451199473487563313551, 6.37528518455242515637822985918, 6.87487728235781384856540194131, 7.81709381647979071707442340991

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