Properties

Label 2-6028-1.1-c1-0-31
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.76·3-s − 2.22·5-s + 2.22·7-s + 0.111·9-s + 11-s + 1.43·13-s − 3.91·15-s − 7.55·17-s − 4.84·19-s + 3.93·21-s + 7.89·23-s − 0.0613·25-s − 5.09·27-s + 3.77·29-s + 10.4·31-s + 1.76·33-s − 4.95·35-s + 7.82·37-s + 2.52·39-s + 6.59·41-s − 5.22·43-s − 0.247·45-s + 1.50·47-s − 2.03·49-s − 13.3·51-s − 5.32·53-s − 2.22·55-s + ⋯
L(s)  = 1  + 1.01·3-s − 0.993·5-s + 0.842·7-s + 0.0371·9-s + 0.301·11-s + 0.397·13-s − 1.01·15-s − 1.83·17-s − 1.11·19-s + 0.857·21-s + 1.64·23-s − 0.0122·25-s − 0.980·27-s + 0.700·29-s + 1.87·31-s + 0.307·33-s − 0.837·35-s + 1.28·37-s + 0.404·39-s + 1.02·41-s − 0.796·43-s − 0.0368·45-s + 0.220·47-s − 0.290·49-s − 1.86·51-s − 0.731·53-s − 0.299·55-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: \(0\)
Selberg data: \((2,\ 6028,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.433828638\)
\(L(\frac12)\) \(\approx\) \(2.433828638\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.76T + 3T^{2} \)
5 \( 1 + 2.22T + 5T^{2} \)
7 \( 1 - 2.22T + 7T^{2} \)
13 \( 1 - 1.43T + 13T^{2} \)
17 \( 1 + 7.55T + 17T^{2} \)
19 \( 1 + 4.84T + 19T^{2} \)
23 \( 1 - 7.89T + 23T^{2} \)
29 \( 1 - 3.77T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 - 7.82T + 37T^{2} \)
41 \( 1 - 6.59T + 41T^{2} \)
43 \( 1 + 5.22T + 43T^{2} \)
47 \( 1 - 1.50T + 47T^{2} \)
53 \( 1 + 5.32T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 5.25T + 61T^{2} \)
67 \( 1 + 2.54T + 67T^{2} \)
71 \( 1 - 4.84T + 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 - 0.949T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−8.181653980070480904632838970735, −7.62879442568076353049070095313, −6.73567574694807569719519286791, −6.16842365741380400676787056209, −4.76238331859898703742244679305, −4.46089686703075705799799190973, −3.65668590719740351052967839034, −2.74214135730530905679995663946, −2.08318777675803370232219511993, −0.77434440161416036497136137327, 0.77434440161416036497136137327, 2.08318777675803370232219511993, 2.74214135730530905679995663946, 3.65668590719740351052967839034, 4.46089686703075705799799190973, 4.76238331859898703742244679305, 6.16842365741380400676787056209, 6.73567574694807569719519286791, 7.62879442568076353049070095313, 8.181653980070480904632838970735

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