Properties

Label 2-6042-1.1-c1-0-21
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 + 0.436·5-s − 6-s − 4.48·7-s − 8-s + 9-s − 0.436·10-s − 1.73·11-s + 12-s + 5.08·13-s + 4.48·14-s + 0.436·15-s + 16-s − 0.656·17-s − 18-s − 19-s + 0.436·20-s − 4.48·21-s + 1.73·22-s + 7.45·23-s − 24-s − 4.80·25-s − 5.08·26-s + 27-s − 4.48·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.195·5-s − 0.408·6-s − 1.69·7-s − 0.353·8-s + 0.333·9-s − 0.138·10-s − 0.523·11-s + 0.288·12-s + 1.41·13-s + 1.19·14-s + 0.112·15-s + 0.250·16-s − 0.159·17-s − 0.235·18-s − 0.229·19-s + 0.0975·20-s − 0.978·21-s + 0.369·22-s + 1.55·23-s − 0.204·24-s − 0.961·25-s − 0.998·26-s + 0.192·27-s − 0.847·28-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

\(L(1)\) \(\approx\) \(1.319526445\)
\(L(\frac12)\) \(\approx\) \(1.319526445\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.436T + 5T^{2} \)
7 \( 1 + 4.48T + 7T^{2} \)
11 \( 1 + 1.73T + 11T^{2} \)
13 \( 1 - 5.08T + 13T^{2} \)
17 \( 1 + 0.656T + 17T^{2} \)
23 \( 1 - 7.45T + 23T^{2} \)
29 \( 1 + 2.33T + 29T^{2} \)
31 \( 1 - 2.69T + 31T^{2} \)
37 \( 1 + 6.70T + 37T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 - 7.23T + 43T^{2} \)
47 \( 1 - 0.322T + 47T^{2} \)
59 \( 1 - 0.584T + 59T^{2} \)
61 \( 1 - 3.23T + 61T^{2} \)
67 \( 1 - 1.52T + 67T^{2} \)
71 \( 1 + 7.62T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 - 6.69T + 79T^{2} \)
83 \( 1 + 4.71T + 83T^{2} \)
89 \( 1 - 8.51T + 89T^{2} \)
97 \( 1 + 4.39T + 97T^{2} \)
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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

−8.244151980278854257880137481110, −7.34904917627634690491420851011, −6.77752152998899483279283178695, −6.13450780894897277804179381694, −5.48013773583491555018604335111, −4.16439603405523283874404439808, −3.34589602675289400209473218060, −2.87378959189165420735242308061, −1.82719573087102497607771249493, −0.64056609324540604316253402698, 0.64056609324540604316253402698, 1.82719573087102497607771249493, 2.87378959189165420735242308061, 3.34589602675289400209473218060, 4.16439603405523283874404439808, 5.48013773583491555018604335111, 6.13450780894897277804179381694, 6.77752152998899483279283178695, 7.34904917627634690491420851011, 8.244151980278854257880137481110

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