Properties

Label 2-6028-1.1-c1-0-33
Degree $2$
Conductor $6028$
Sign $-1$
Analytic cond. $48.1338$
Root an. cond. $6.93785$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.30·3-s − 2.25·5-s − 2.39·7-s + 2.30·9-s + 11-s − 2.79·13-s + 5.19·15-s + 1.45·17-s − 6.96·19-s + 5.51·21-s + 2.28·23-s + 0.0906·25-s + 1.60·27-s + 4.69·29-s − 2.86·31-s − 2.30·33-s + 5.39·35-s + 5.35·37-s + 6.44·39-s + 1.80·41-s + 5.80·43-s − 5.19·45-s + 5.45·47-s − 1.27·49-s − 3.36·51-s + 3.77·53-s − 2.25·55-s + ⋯
L(s)  = 1  − 1.32·3-s − 1.00·5-s − 0.904·7-s + 0.767·9-s + 0.301·11-s − 0.776·13-s + 1.34·15-s + 0.354·17-s − 1.59·19-s + 1.20·21-s + 0.476·23-s + 0.0181·25-s + 0.309·27-s + 0.872·29-s − 0.513·31-s − 0.400·33-s + 0.912·35-s + 0.881·37-s + 1.03·39-s + 0.282·41-s + 0.885·43-s − 0.774·45-s + 0.796·47-s − 0.181·49-s − 0.470·51-s + 0.518·53-s − 0.304·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6028\)    =    \(2^{2} \cdot 11 \cdot 137\)
Sign: $-1$
Analytic conductor: \(48.1338\)
Root analytic conductor: \(6.93785\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6028,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 - T \)
137 \( 1 + T \)
good3 \( 1 + 2.30T + 3T^{2} \)
5 \( 1 + 2.25T + 5T^{2} \)
7 \( 1 + 2.39T + 7T^{2} \)
13 \( 1 + 2.79T + 13T^{2} \)
17 \( 1 - 1.45T + 17T^{2} \)
19 \( 1 + 6.96T + 19T^{2} \)
23 \( 1 - 2.28T + 23T^{2} \)
29 \( 1 - 4.69T + 29T^{2} \)
31 \( 1 + 2.86T + 31T^{2} \)
37 \( 1 - 5.35T + 37T^{2} \)
41 \( 1 - 1.80T + 41T^{2} \)
43 \( 1 - 5.80T + 43T^{2} \)
47 \( 1 - 5.45T + 47T^{2} \)
53 \( 1 - 3.77T + 53T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 0.632T + 61T^{2} \)
67 \( 1 + 8.66T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 6.97T + 73T^{2} \)
79 \( 1 - 4.41T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 + 0.855T + 97T^{2} \)
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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

−7.43826406222163176842107422366, −7.01472117764977679856740518439, −6.17707446069627460582448402221, −5.80126740018601944609952056761, −4.69869764698034429971578802315, −4.28069248226960069615380681885, −3.35814576622199631288939846630, −2.37408046162847524022008354528, −0.817959950311494957819764603350, 0, 0.817959950311494957819764603350, 2.37408046162847524022008354528, 3.35814576622199631288939846630, 4.28069248226960069615380681885, 4.69869764698034429971578802315, 5.80126740018601944609952056761, 6.17707446069627460582448402221, 7.01472117764977679856740518439, 7.43826406222163176842107422366

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