Properties

Label 2-6027-1.1-c1-0-261
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  + 2.14·2-s − 3-s + 2.61·4-s + 1.12·5-s − 2.14·6-s + 1.32·8-s + 9-s + 2.41·10-s + 5.38·11-s − 2.61·12-s − 5.09·13-s − 1.12·15-s − 2.38·16-s − 4.12·17-s + 2.14·18-s − 2.31·19-s + 2.93·20-s + 11.5·22-s − 7.27·23-s − 1.32·24-s − 3.74·25-s − 10.9·26-s − 27-s + 1.41·29-s − 2.41·30-s − 1.68·31-s − 7.77·32-s + ⋯
L(s)  = 1  + 1.51·2-s − 0.577·3-s + 1.30·4-s + 0.501·5-s − 0.877·6-s + 0.469·8-s + 0.333·9-s + 0.762·10-s + 1.62·11-s − 0.755·12-s − 1.41·13-s − 0.289·15-s − 0.595·16-s − 0.999·17-s + 0.506·18-s − 0.531·19-s + 0.657·20-s + 2.46·22-s − 1.51·23-s − 0.271·24-s − 0.748·25-s − 2.14·26-s − 0.192·27-s + 0.262·29-s − 0.440·30-s − 0.302·31-s − 1.37·32-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 \( 1 - 2.14T + 2T^{2} \)
5 \( 1 - 1.12T + 5T^{2} \)
11 \( 1 - 5.38T + 11T^{2} \)
13 \( 1 + 5.09T + 13T^{2} \)
17 \( 1 + 4.12T + 17T^{2} \)
19 \( 1 + 2.31T + 19T^{2} \)
23 \( 1 + 7.27T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 1.68T + 31T^{2} \)
37 \( 1 + 6.86T + 37T^{2} \)
43 \( 1 + 1.11T + 43T^{2} \)
47 \( 1 - 1.87T + 47T^{2} \)
53 \( 1 + 8.32T + 53T^{2} \)
59 \( 1 + 6.42T + 59T^{2} \)
61 \( 1 - 14.9T + 61T^{2} \)
67 \( 1 + 6.69T + 67T^{2} \)
71 \( 1 - 2.41T + 71T^{2} \)
73 \( 1 - 8.85T + 73T^{2} \)
79 \( 1 - 1.36T + 79T^{2} \)
83 \( 1 - 5.35T + 83T^{2} \)
89 \( 1 + 3.08T + 89T^{2} \)
97 \( 1 + 9.41T + 97T^{2} \)
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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

−7.26058981618342951598938493035, −6.64300252045573897227685997967, −6.21204596899562961230553248022, −5.53542520525175344202591196693, −4.75232904080527185088437523496, −4.20396291716412772201040066009, −3.57862782909946700058056215517, −2.33813419254822556875654056198, −1.79014450543595923552451435921, 0, 1.79014450543595923552451435921, 2.33813419254822556875654056198, 3.57862782909946700058056215517, 4.20396291716412772201040066009, 4.75232904080527185088437523496, 5.53542520525175344202591196693, 6.21204596899562961230553248022, 6.64300252045573897227685997967, 7.26058981618342951598938493035

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