Properties

Label 2-6042-1.1-c1-0-36
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 − 4.29·5-s + 6-s + 4.11·7-s + 8-s + 9-s − 4.29·10-s − 1.82·11-s + 12-s + 0.829·13-s + 4.11·14-s − 4.29·15-s + 16-s − 5.01·17-s + 18-s + 19-s − 4.29·20-s + 4.11·21-s − 1.82·22-s + 0.0570·23-s + 24-s + 13.4·25-s + 0.829·26-s + 27-s + 4.11·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.91·5-s + 0.408·6-s + 1.55·7-s + 0.353·8-s + 0.333·9-s − 1.35·10-s − 0.550·11-s + 0.288·12-s + 0.229·13-s + 1.09·14-s − 1.10·15-s + 0.250·16-s − 1.21·17-s + 0.235·18-s + 0.229·19-s − 0.959·20-s + 0.898·21-s − 0.388·22-s + 0.0119·23-s + 0.204·24-s + 2.68·25-s + 0.162·26-s + 0.192·27-s + 0.777·28-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

\(L(1)\) \(\approx\) \(3.188796670\)
\(L(\frac12)\) \(\approx\) \(3.188796670\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.29T + 5T^{2} \)
7 \( 1 - 4.11T + 7T^{2} \)
11 \( 1 + 1.82T + 11T^{2} \)
13 \( 1 - 0.829T + 13T^{2} \)
17 \( 1 + 5.01T + 17T^{2} \)
23 \( 1 - 0.0570T + 23T^{2} \)
29 \( 1 - 1.24T + 29T^{2} \)
31 \( 1 + 1.49T + 31T^{2} \)
37 \( 1 + 3.66T + 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 - 7.56T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
59 \( 1 + 5.60T + 59T^{2} \)
61 \( 1 - 9.46T + 61T^{2} \)
67 \( 1 + 5.88T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 + 5.81T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 - 0.609T + 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 0.790T + 97T^{2} \)
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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

−7.77240924647120875540499533683, −7.62097490837954069660984497246, −6.94195487667335812162101407681, −5.78034990520347972051259344055, −4.79471450796242666179337683203, −4.40839782347360370863692847050, −3.86350143638103799943792768995, −2.92392029267560172390719752362, −2.09539654815201861366200395790, −0.827405543399686897923846245869, 0.827405543399686897923846245869, 2.09539654815201861366200395790, 2.92392029267560172390719752362, 3.86350143638103799943792768995, 4.40839782347360370863692847050, 4.79471450796242666179337683203, 5.78034990520347972051259344055, 6.94195487667335812162101407681, 7.62097490837954069660984497246, 7.77240924647120875540499533683

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