Properties

Label 2-6027-1.1-c1-0-132
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  − 1.21·2-s + 3-s − 0.518·4-s − 3.45·5-s − 1.21·6-s + 3.06·8-s + 9-s + 4.20·10-s − 4.83·11-s − 0.518·12-s + 4.46·13-s − 3.45·15-s − 2.69·16-s − 3.17·17-s − 1.21·18-s − 1.25·19-s + 1.79·20-s + 5.87·22-s + 3.08·23-s + 3.06·24-s + 6.91·25-s − 5.43·26-s + 27-s − 5.78·29-s + 4.20·30-s + 8.75·31-s − 2.85·32-s + ⋯
L(s)  = 1  − 0.860·2-s + 0.577·3-s − 0.259·4-s − 1.54·5-s − 0.496·6-s + 1.08·8-s + 0.333·9-s + 1.32·10-s − 1.45·11-s − 0.149·12-s + 1.23·13-s − 0.891·15-s − 0.673·16-s − 0.770·17-s − 0.286·18-s − 0.288·19-s + 0.400·20-s + 1.25·22-s + 0.642·23-s + 0.625·24-s + 1.38·25-s − 1.06·26-s + 0.192·27-s − 1.07·29-s + 0.766·30-s + 1.57·31-s − 0.504·32-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 \( 1 + 1.21T + 2T^{2} \)
5 \( 1 + 3.45T + 5T^{2} \)
11 \( 1 + 4.83T + 11T^{2} \)
13 \( 1 - 4.46T + 13T^{2} \)
17 \( 1 + 3.17T + 17T^{2} \)
19 \( 1 + 1.25T + 19T^{2} \)
23 \( 1 - 3.08T + 23T^{2} \)
29 \( 1 + 5.78T + 29T^{2} \)
31 \( 1 - 8.75T + 31T^{2} \)
37 \( 1 + 1.31T + 37T^{2} \)
43 \( 1 + 7.25T + 43T^{2} \)
47 \( 1 - 5.70T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 7.77T + 59T^{2} \)
61 \( 1 + 3.37T + 61T^{2} \)
67 \( 1 - 8.54T + 67T^{2} \)
71 \( 1 - 7.81T + 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 - 6.38T + 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 - 0.678T + 97T^{2} \)
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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

−8.023440581546460202413642690833, −7.34406381568550270184722005949, −6.70851476989172227916116714204, −5.44541605151728267434757747408, −4.60396986038580739776608844185, −4.01536592721106406924976020690, −3.28802885811216984193979311828, −2.28171617549182355746022570209, −0.982403273887423579257546954887, 0, 0.982403273887423579257546954887, 2.28171617549182355746022570209, 3.28802885811216984193979311828, 4.01536592721106406924976020690, 4.60396986038580739776608844185, 5.44541605151728267434757747408, 6.70851476989172227916116714204, 7.34406381568550270184722005949, 8.023440581546460202413642690833

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