Properties

Label 2-6016-1.1-c1-0-76
Degree $2$
Conductor $6016$
Sign $1$
Analytic cond. $48.0380$
Root an. cond. $6.93094$
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  + 0.904·3-s + 3.10·5-s + 1.70·7-s − 2.18·9-s − 6.05·11-s + 1.97·13-s + 2.80·15-s + 0.802·17-s + 2.24·19-s + 1.54·21-s − 2.79·23-s + 4.63·25-s − 4.68·27-s + 9.23·29-s + 7.61·31-s − 5.48·33-s + 5.30·35-s + 3.25·37-s + 1.78·39-s + 3.62·41-s − 0.635·43-s − 6.76·45-s + 47-s − 4.08·49-s + 0.725·51-s − 4.24·53-s − 18.8·55-s + ⋯
L(s)  = 1  + 0.522·3-s + 1.38·5-s + 0.645·7-s − 0.727·9-s − 1.82·11-s + 0.547·13-s + 0.725·15-s + 0.194·17-s + 0.515·19-s + 0.337·21-s − 0.581·23-s + 0.926·25-s − 0.902·27-s + 1.71·29-s + 1.36·31-s − 0.954·33-s + 0.896·35-s + 0.535·37-s + 0.286·39-s + 0.566·41-s − 0.0969·43-s − 1.00·45-s + 0.145·47-s − 0.583·49-s + 0.101·51-s − 0.583·53-s − 2.53·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6016\)    =    \(2^{7} \cdot 47\)
Sign: $1$
Analytic conductor: \(48.0380\)
Root analytic conductor: \(6.93094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.126290033\)
\(L(\frac12)\) \(\approx\) \(3.126290033\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
47 \( 1 - T \)
good3 \( 1 - 0.904T + 3T^{2} \)
5 \( 1 - 3.10T + 5T^{2} \)
7 \( 1 - 1.70T + 7T^{2} \)
11 \( 1 + 6.05T + 11T^{2} \)
13 \( 1 - 1.97T + 13T^{2} \)
17 \( 1 - 0.802T + 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
23 \( 1 + 2.79T + 23T^{2} \)
29 \( 1 - 9.23T + 29T^{2} \)
31 \( 1 - 7.61T + 31T^{2} \)
37 \( 1 - 3.25T + 37T^{2} \)
41 \( 1 - 3.62T + 41T^{2} \)
43 \( 1 + 0.635T + 43T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 - 2.01T + 59T^{2} \)
61 \( 1 - 6.89T + 61T^{2} \)
67 \( 1 + 6.92T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 14.6T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 0.818T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−8.193264235851538221109555700600, −7.64743568347244483574892750175, −6.44914333295194108108501823905, −5.92988090833364857783025397080, −5.21186914448418107208437045050, −4.72073425236610297579065666468, −3.36304126778614152900032357241, −2.57137436320194031732318087171, −2.15337934920042101677154663481, −0.909708294562438124989072847372, 0.909708294562438124989072847372, 2.15337934920042101677154663481, 2.57137436320194031732318087171, 3.36304126778614152900032357241, 4.72073425236610297579065666468, 5.21186914448418107208437045050, 5.92988090833364857783025397080, 6.44914333295194108108501823905, 7.64743568347244483574892750175, 8.193264235851538221109555700600

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