Properties

Label 2-6027-1.1-c1-0-5
Degree $2$
Conductor $6027$
Sign $1$
Analytic cond. $48.1258$
Root an. cond. $6.93727$
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  − 1.44·2-s − 3-s + 0.0982·4-s − 3.93·5-s + 1.44·6-s + 2.75·8-s + 9-s + 5.69·10-s − 1.25·11-s − 0.0982·12-s − 0.0627·13-s + 3.93·15-s − 4.18·16-s − 6.18·17-s − 1.44·18-s + 0.467·19-s − 0.386·20-s + 1.81·22-s − 4.63·23-s − 2.75·24-s + 10.4·25-s + 0.0908·26-s − 27-s + 0.429·29-s − 5.69·30-s − 2.26·31-s + 0.555·32-s + ⋯
L(s)  = 1  − 1.02·2-s − 0.577·3-s + 0.0491·4-s − 1.75·5-s + 0.591·6-s + 0.973·8-s + 0.333·9-s + 1.80·10-s − 0.378·11-s − 0.0283·12-s − 0.0173·13-s + 1.01·15-s − 1.04·16-s − 1.50·17-s − 0.341·18-s + 0.107·19-s − 0.0864·20-s + 0.387·22-s − 0.966·23-s − 0.562·24-s + 2.09·25-s + 0.0178·26-s − 0.192·27-s + 0.0797·29-s − 1.03·30-s − 0.405·31-s + 0.0981·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(48.1258\)
Root analytic conductor: \(6.93727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6027,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05549366639\)
\(L(\frac12)\) \(\approx\) \(0.05549366639\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 + 1.44T + 2T^{2} \)
5 \( 1 + 3.93T + 5T^{2} \)
11 \( 1 + 1.25T + 11T^{2} \)
13 \( 1 + 0.0627T + 13T^{2} \)
17 \( 1 + 6.18T + 17T^{2} \)
19 \( 1 - 0.467T + 19T^{2} \)
23 \( 1 + 4.63T + 23T^{2} \)
29 \( 1 - 0.429T + 29T^{2} \)
31 \( 1 + 2.26T + 31T^{2} \)
37 \( 1 + 1.50T + 37T^{2} \)
43 \( 1 - 6.63T + 43T^{2} \)
47 \( 1 - 2.59T + 47T^{2} \)
53 \( 1 + 3.82T + 53T^{2} \)
59 \( 1 - 1.80T + 59T^{2} \)
61 \( 1 + 8.03T + 61T^{2} \)
67 \( 1 + 0.159T + 67T^{2} \)
71 \( 1 + 7.58T + 71T^{2} \)
73 \( 1 + 9.71T + 73T^{2} \)
79 \( 1 + 5.50T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + 9.15T + 89T^{2} \)
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

−8.189149357847470836459359517695, −7.37173533368533766850664888912, −7.12648858403659371186450194058, −6.10513141668515424344803088340, −5.03563024299324015657028814423, −4.30409930853687224679702734486, −3.95926673777151065504627689190, −2.68364752630669583111738765810, −1.41474242910462077956052261915, −0.15795818972150154622611661394, 0.15795818972150154622611661394, 1.41474242910462077956052261915, 2.68364752630669583111738765810, 3.95926673777151065504627689190, 4.30409930853687224679702734486, 5.03563024299324015657028814423, 6.10513141668515424344803088340, 7.12648858403659371186450194058, 7.37173533368533766850664888912, 8.189149357847470836459359517695

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