Properties

Label 2-6042-1.1-c1-0-10
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 − 2.67·5-s − 6-s + 2.79·7-s + 8-s + 9-s − 2.67·10-s − 4.92·11-s − 12-s − 6.47·13-s + 2.79·14-s + 2.67·15-s + 16-s + 2·17-s + 18-s − 19-s − 2.67·20-s − 2.79·21-s − 4.92·22-s − 4.24·23-s − 24-s + 2.15·25-s − 6.47·26-s − 27-s + 2.79·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.19·5-s − 0.408·6-s + 1.05·7-s + 0.353·8-s + 0.333·9-s − 0.846·10-s − 1.48·11-s − 0.288·12-s − 1.79·13-s + 0.746·14-s + 0.690·15-s + 0.250·16-s + 0.485·17-s + 0.235·18-s − 0.229·19-s − 0.598·20-s − 0.609·21-s − 1.04·22-s − 0.885·23-s − 0.204·24-s + 0.431·25-s − 1.26·26-s − 0.192·27-s + 0.527·28-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

\(L(1)\) \(\approx\) \(1.420116201\)
\(L(\frac12)\) \(\approx\) \(1.420116201\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.67T + 5T^{2} \)
7 \( 1 - 2.79T + 7T^{2} \)
11 \( 1 + 4.92T + 11T^{2} \)
13 \( 1 + 6.47T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
23 \( 1 + 4.24T + 23T^{2} \)
29 \( 1 - 6.24T + 29T^{2} \)
31 \( 1 + 2.73T + 31T^{2} \)
37 \( 1 - 1.11T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 + 5.71T + 43T^{2} \)
47 \( 1 - 5.20T + 47T^{2} \)
59 \( 1 - 0.150T + 59T^{2} \)
61 \( 1 - 9.72T + 61T^{2} \)
67 \( 1 - 6.79T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 4.26T + 73T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−7.79271110974461028278797309984, −7.54940051554267481618418847387, −6.71835464507516490055942680120, −5.62251831017561041738795178112, −5.08305343327476778237127662947, −4.59128999990262245631563240691, −3.91048837253194134815910176999, −2.77895714458371651665794576954, −2.08486854872079991857491740535, −0.55226727858622188062498821173, 0.55226727858622188062498821173, 2.08486854872079991857491740535, 2.77895714458371651665794576954, 3.91048837253194134815910176999, 4.59128999990262245631563240691, 5.08305343327476778237127662947, 5.62251831017561041738795178112, 6.71835464507516490055942680120, 7.54940051554267481618418847387, 7.79271110974461028278797309984

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