Properties

Label 2-6027-1.1-c1-0-227
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  − 2.20·2-s + 3-s + 2.84·4-s + 2.77·5-s − 2.20·6-s − 1.85·8-s + 9-s − 6.11·10-s − 1.68·11-s + 2.84·12-s + 1.26·13-s + 2.77·15-s − 1.59·16-s − 5.32·17-s − 2.20·18-s − 0.141·19-s + 7.89·20-s + 3.69·22-s − 3.48·23-s − 1.85·24-s + 2.70·25-s − 2.78·26-s + 27-s − 2.99·29-s − 6.11·30-s − 5.66·31-s + 7.23·32-s + ⋯
L(s)  = 1  − 1.55·2-s + 0.577·3-s + 1.42·4-s + 1.24·5-s − 0.898·6-s − 0.657·8-s + 0.333·9-s − 1.93·10-s − 0.506·11-s + 0.821·12-s + 0.351·13-s + 0.716·15-s − 0.399·16-s − 1.29·17-s − 0.518·18-s − 0.0325·19-s + 1.76·20-s + 0.788·22-s − 0.725·23-s − 0.379·24-s + 0.541·25-s − 0.547·26-s + 0.192·27-s − 0.555·29-s − 1.11·30-s − 1.01·31-s + 1.27·32-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 \( 1 + 2.20T + 2T^{2} \)
5 \( 1 - 2.77T + 5T^{2} \)
11 \( 1 + 1.68T + 11T^{2} \)
13 \( 1 - 1.26T + 13T^{2} \)
17 \( 1 + 5.32T + 17T^{2} \)
19 \( 1 + 0.141T + 19T^{2} \)
23 \( 1 + 3.48T + 23T^{2} \)
29 \( 1 + 2.99T + 29T^{2} \)
31 \( 1 + 5.66T + 31T^{2} \)
37 \( 1 - 0.881T + 37T^{2} \)
43 \( 1 - 4.77T + 43T^{2} \)
47 \( 1 - 6.78T + 47T^{2} \)
53 \( 1 + 9.07T + 53T^{2} \)
59 \( 1 + 3.92T + 59T^{2} \)
61 \( 1 + 9.47T + 61T^{2} \)
67 \( 1 - 5.95T + 67T^{2} \)
71 \( 1 + 2.01T + 71T^{2} \)
73 \( 1 + 1.94T + 73T^{2} \)
79 \( 1 + 9.24T + 79T^{2} \)
83 \( 1 + 0.172T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 - 5.03T + 97T^{2} \)
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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

−7.79493754261778896833396345575, −7.36191768720554546382265470474, −6.43053994744484053405184709332, −5.93310167901267228377523779167, −4.89588495741301954610446309811, −3.92629620075795707982191544522, −2.65851797854198838670715461835, −2.08976157615946192173990072566, −1.41743668688711724449810512499, 0, 1.41743668688711724449810512499, 2.08976157615946192173990072566, 2.65851797854198838670715461835, 3.92629620075795707982191544522, 4.89588495741301954610446309811, 5.93310167901267228377523779167, 6.43053994744484053405184709332, 7.36191768720554546382265470474, 7.79493754261778896833396345575

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