Properties

Label 2-6042-1.1-c1-0-100
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 + 3.88·5-s − 6-s − 0.785·7-s + 8-s + 9-s + 3.88·10-s + 4.53·11-s − 12-s + 6.52·13-s − 0.785·14-s − 3.88·15-s + 16-s + 2·17-s + 18-s − 19-s + 3.88·20-s + 0.785·21-s + 4.53·22-s − 1.34·23-s − 24-s + 10.1·25-s + 6.52·26-s − 27-s − 0.785·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.73·5-s − 0.408·6-s − 0.296·7-s + 0.353·8-s + 0.333·9-s + 1.22·10-s + 1.36·11-s − 0.288·12-s + 1.81·13-s − 0.209·14-s − 1.00·15-s + 0.250·16-s + 0.485·17-s + 0.235·18-s − 0.229·19-s + 0.869·20-s + 0.171·21-s + 0.967·22-s − 0.281·23-s − 0.204·24-s + 2.02·25-s + 1.28·26-s − 0.192·27-s − 0.148·28-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

\(L(1)\) \(\approx\) \(4.636004324\)
\(L(\frac12)\) \(\approx\) \(4.636004324\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.88T + 5T^{2} \)
7 \( 1 + 0.785T + 7T^{2} \)
11 \( 1 - 4.53T + 11T^{2} \)
13 \( 1 - 6.52T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
23 \( 1 + 1.34T + 23T^{2} \)
29 \( 1 - 3.34T + 29T^{2} \)
31 \( 1 - 0.236T + 31T^{2} \)
37 \( 1 - 1.24T + 37T^{2} \)
41 \( 1 - 2.68T + 41T^{2} \)
43 \( 1 + 7.25T + 43T^{2} \)
47 \( 1 + 2.65T + 47T^{2} \)
59 \( 1 + 2.72T + 59T^{2} \)
61 \( 1 + 1.81T + 61T^{2} \)
67 \( 1 + 2.50T + 67T^{2} \)
71 \( 1 + 6.79T + 71T^{2} \)
73 \( 1 - 3.73T + 73T^{2} \)
79 \( 1 + 7.44T + 79T^{2} \)
83 \( 1 - 3.60T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 7.66T + 97T^{2} \)
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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

−8.099518658628594779646409689546, −6.73952406757450715577763383965, −6.50799224957048613488400496766, −5.95622904390597429121355767089, −5.45585123407634888235413760433, −4.50145100415842077449015227808, −3.71116122281191047627119257837, −2.87214215823742210825171056501, −1.62674994388021305431788481770, −1.25783968468931333860484642273, 1.25783968468931333860484642273, 1.62674994388021305431788481770, 2.87214215823742210825171056501, 3.71116122281191047627119257837, 4.50145100415842077449015227808, 5.45585123407634888235413760433, 5.95622904390597429121355767089, 6.50799224957048613488400496766, 6.73952406757450715577763383965, 8.099518658628594779646409689546

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