Properties

Label 2-6027-1.1-c1-0-121
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  + 0.697·2-s − 3-s − 1.51·4-s − 3.10·5-s − 0.697·6-s − 2.45·8-s + 9-s − 2.16·10-s + 1.68·11-s + 1.51·12-s − 1.74·13-s + 3.10·15-s + 1.31·16-s + 0.100·17-s + 0.697·18-s − 4.73·19-s + 4.69·20-s + 1.17·22-s + 6.23·23-s + 2.45·24-s + 4.61·25-s − 1.21·26-s − 27-s − 6.80·29-s + 2.16·30-s + 0.735·31-s + 5.82·32-s + ⋯
L(s)  = 1  + 0.493·2-s − 0.577·3-s − 0.756·4-s − 1.38·5-s − 0.284·6-s − 0.866·8-s + 0.333·9-s − 0.684·10-s + 0.507·11-s + 0.436·12-s − 0.484·13-s + 0.800·15-s + 0.329·16-s + 0.0244·17-s + 0.164·18-s − 1.08·19-s + 1.04·20-s + 0.250·22-s + 1.29·23-s + 0.500·24-s + 0.922·25-s − 0.238·26-s − 0.192·27-s − 1.26·29-s + 0.394·30-s + 0.132·31-s + 1.02·32-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 \( 1 - 0.697T + 2T^{2} \)
5 \( 1 + 3.10T + 5T^{2} \)
11 \( 1 - 1.68T + 11T^{2} \)
13 \( 1 + 1.74T + 13T^{2} \)
17 \( 1 - 0.100T + 17T^{2} \)
19 \( 1 + 4.73T + 19T^{2} \)
23 \( 1 - 6.23T + 23T^{2} \)
29 \( 1 + 6.80T + 29T^{2} \)
31 \( 1 - 0.735T + 31T^{2} \)
37 \( 1 - 5.71T + 37T^{2} \)
43 \( 1 - 2.92T + 43T^{2} \)
47 \( 1 - 9.74T + 47T^{2} \)
53 \( 1 - 3.28T + 53T^{2} \)
59 \( 1 - 0.705T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 + 5.41T + 67T^{2} \)
71 \( 1 + 8.17T + 71T^{2} \)
73 \( 1 + 0.0702T + 73T^{2} \)
79 \( 1 - 9.33T + 79T^{2} \)
83 \( 1 + 7.84T + 83T^{2} \)
89 \( 1 + 2.28T + 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−7.60225729730594897710799552683, −7.07460844910018380288770942043, −6.19171705076719063315646082030, −5.44145227014154075459982829928, −4.66933086242759725527899102567, −4.11672226685035845174033175568, −3.62700659588397528587789714020, −2.54974235847051188771886601034, −0.943769022431447146803064647171, 0, 0.943769022431447146803064647171, 2.54974235847051188771886601034, 3.62700659588397528587789714020, 4.11672226685035845174033175568, 4.66933086242759725527899102567, 5.44145227014154075459982829928, 6.19171705076719063315646082030, 7.07460844910018380288770942043, 7.60225729730594897710799552683

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