Properties

Label 2-6042-1.1-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 2.83·5-s + 6-s − 4.03·7-s − 8-s + 9-s + 2.83·10-s + 1.63·11-s − 12-s + 2.83·13-s + 4.03·14-s + 2.83·15-s + 16-s + 6.03·17-s − 18-s + 19-s − 2.83·20-s + 4.03·21-s − 1.63·22-s + 5.46·23-s + 24-s + 3.03·25-s − 2.83·26-s − 27-s − 4.03·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.26·5-s + 0.408·6-s − 1.52·7-s − 0.353·8-s + 0.333·9-s + 0.896·10-s + 0.493·11-s − 0.288·12-s + 0.786·13-s + 1.07·14-s + 0.731·15-s + 0.250·16-s + 1.46·17-s − 0.235·18-s + 0.229·19-s − 0.633·20-s + 0.880·21-s − 0.348·22-s + 1.14·23-s + 0.204·24-s + 0.606·25-s − 0.555·26-s − 0.192·27-s − 0.762·28-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: \(0\)
Selberg data: \((2,\ 6042,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5358906521\)
\(L(\frac12)\) \(\approx\) \(0.5358906521\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.83T + 5T^{2} \)
7 \( 1 + 4.03T + 7T^{2} \)
11 \( 1 - 1.63T + 11T^{2} \)
13 \( 1 - 2.83T + 13T^{2} \)
17 \( 1 - 6.03T + 17T^{2} \)
23 \( 1 - 5.46T + 23T^{2} \)
29 \( 1 + 8.62T + 29T^{2} \)
31 \( 1 - 1.27T + 31T^{2} \)
37 \( 1 - 3.63T + 37T^{2} \)
41 \( 1 + 0.867T + 41T^{2} \)
43 \( 1 + 5.46T + 43T^{2} \)
47 \( 1 + 7.39T + 47T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 - 0.768T + 67T^{2} \)
71 \( 1 - 1.60T + 71T^{2} \)
73 \( 1 + 0.761T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 + 8.70T + 89T^{2} \)
97 \( 1 + 9.13T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.031143165374386301761376939637, −7.29796671364614912397572497093, −6.85334179343506122103541039265, −6.07317935469097713842482463199, −5.44994343887715437629214696224, −4.24181121667867412414242092575, −3.43829666412722359614515297414, −3.11580301221858800608677010678, −1.41643573621987703465073744020, −0.47307858427833738323096286378, 0.47307858427833738323096286378, 1.41643573621987703465073744020, 3.11580301221858800608677010678, 3.43829666412722359614515297414, 4.24181121667867412414242092575, 5.44994343887715437629214696224, 6.07317935469097713842482463199, 6.85334179343506122103541039265, 7.29796671364614912397572497093, 8.031143165374386301761376939637

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