Properties

Label 2-6027-1.1-c1-0-244
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.48·2-s + 3-s + 0.193·4-s − 0.193·5-s + 1.48·6-s − 2.67·8-s + 9-s − 0.287·10-s − 5.35·11-s + 0.193·12-s + 3.86·13-s − 0.193·15-s − 4.35·16-s + 0.156·17-s + 1.48·18-s + 1.84·19-s − 0.0376·20-s − 7.92·22-s + 7.79·23-s − 2.67·24-s − 4.96·25-s + 5.73·26-s + 27-s − 4.28·29-s − 0.287·30-s − 0.806·31-s − 1.09·32-s + ⋯
L(s)  = 1  + 1.04·2-s + 0.577·3-s + 0.0969·4-s − 0.0867·5-s + 0.604·6-s − 0.945·8-s + 0.333·9-s − 0.0908·10-s − 1.61·11-s + 0.0559·12-s + 1.07·13-s − 0.0500·15-s − 1.08·16-s + 0.0379·17-s + 0.349·18-s + 0.422·19-s − 0.00841·20-s − 1.68·22-s + 1.62·23-s − 0.546·24-s − 0.992·25-s + 1.12·26-s + 0.192·27-s − 0.796·29-s − 0.0524·30-s − 0.144·31-s − 0.193·32-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: \(1\)
Selberg data: \((2,\ 6027,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 - 1.48T + 2T^{2} \)
5 \( 1 + 0.193T + 5T^{2} \)
11 \( 1 + 5.35T + 11T^{2} \)
13 \( 1 - 3.86T + 13T^{2} \)
17 \( 1 - 0.156T + 17T^{2} \)
19 \( 1 - 1.84T + 19T^{2} \)
23 \( 1 - 7.79T + 23T^{2} \)
29 \( 1 + 4.28T + 29T^{2} \)
31 \( 1 + 0.806T + 31T^{2} \)
37 \( 1 + 5.80T + 37T^{2} \)
43 \( 1 + 6.41T + 43T^{2} \)
47 \( 1 - 3.15T + 47T^{2} \)
53 \( 1 + 1.86T + 53T^{2} \)
59 \( 1 - 1.96T + 59T^{2} \)
61 \( 1 + 6.15T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + 1.61T + 79T^{2} \)
83 \( 1 + 6.12T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 - 2.28T + 97T^{2} \)
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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

−7.63183898751018016689072145684, −7.06196290547570727650769735785, −6.00967013636130482670364535497, −5.46307426985463261693863741140, −4.83043003382826945721774196267, −4.00911457912532260137895134282, −3.21934928339214477170660564004, −2.79510225462148964021179757541, −1.57880351898408092494449828744, 0, 1.57880351898408092494449828744, 2.79510225462148964021179757541, 3.21934928339214477170660564004, 4.00911457912532260137895134282, 4.83043003382826945721774196267, 5.46307426985463261693863741140, 6.00967013636130482670364535497, 7.06196290547570727650769735785, 7.63183898751018016689072145684

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