Properties

Label 2-6027-1.1-c1-0-211
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  − 2.73·2-s + 3-s + 5.47·4-s + 2.29·5-s − 2.73·6-s − 9.49·8-s + 9-s − 6.26·10-s − 0.920·11-s + 5.47·12-s − 6.70·13-s + 2.29·15-s + 15.0·16-s + 3.80·17-s − 2.73·18-s + 6.56·19-s + 12.5·20-s + 2.51·22-s − 6.79·23-s − 9.49·24-s + 0.244·25-s + 18.3·26-s + 27-s − 4.96·29-s − 6.26·30-s + 1.14·31-s − 22.0·32-s + ⋯
L(s)  = 1  − 1.93·2-s + 0.577·3-s + 2.73·4-s + 1.02·5-s − 1.11·6-s − 3.35·8-s + 0.333·9-s − 1.97·10-s − 0.277·11-s + 1.58·12-s − 1.86·13-s + 0.591·15-s + 3.75·16-s + 0.923·17-s − 0.644·18-s + 1.50·19-s + 2.80·20-s + 0.536·22-s − 1.41·23-s − 1.93·24-s + 0.0489·25-s + 3.59·26-s + 0.192·27-s − 0.921·29-s − 1.14·30-s + 0.205·31-s − 3.90·32-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 \( 1 + 2.73T + 2T^{2} \)
5 \( 1 - 2.29T + 5T^{2} \)
11 \( 1 + 0.920T + 11T^{2} \)
13 \( 1 + 6.70T + 13T^{2} \)
17 \( 1 - 3.80T + 17T^{2} \)
19 \( 1 - 6.56T + 19T^{2} \)
23 \( 1 + 6.79T + 23T^{2} \)
29 \( 1 + 4.96T + 29T^{2} \)
31 \( 1 - 1.14T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
43 \( 1 - 3.89T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 7.81T + 53T^{2} \)
59 \( 1 - 0.168T + 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 - 1.48T + 67T^{2} \)
71 \( 1 + 6.79T + 71T^{2} \)
73 \( 1 - 4.20T + 73T^{2} \)
79 \( 1 - 4.67T + 79T^{2} \)
83 \( 1 + 6.73T + 83T^{2} \)
89 \( 1 + 8.78T + 89T^{2} \)
97 \( 1 - 1.94T + 97T^{2} \)
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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

−7.83258758240338814275732289191, −7.31639360459706657849341836325, −6.75969076925119733245730457778, −5.68055809530208843514586809893, −5.26878065560471884469732828448, −3.58894478372262463224794361459, −2.65881603421315963689920104401, −2.11880884219193505057339618633, −1.34521799157805738551496261810, 0, 1.34521799157805738551496261810, 2.11880884219193505057339618633, 2.65881603421315963689920104401, 3.58894478372262463224794361459, 5.26878065560471884469732828448, 5.68055809530208843514586809893, 6.75969076925119733245730457778, 7.31639360459706657849341836325, 7.83258758240338814275732289191

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