Properties

Label 2-6027-1.1-c1-0-264
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  + 2.47·2-s + 3-s + 4.11·4-s − 4.25·5-s + 2.47·6-s + 5.22·8-s + 9-s − 10.5·10-s − 0.0562·11-s + 4.11·12-s − 0.121·13-s − 4.25·15-s + 4.69·16-s − 6.50·17-s + 2.47·18-s + 2.99·19-s − 17.5·20-s − 0.139·22-s − 7.44·23-s + 5.22·24-s + 13.1·25-s − 0.300·26-s + 27-s + 6.83·29-s − 10.5·30-s − 5.89·31-s + 1.15·32-s + ⋯
L(s)  = 1  + 1.74·2-s + 0.577·3-s + 2.05·4-s − 1.90·5-s + 1.00·6-s + 1.84·8-s + 0.333·9-s − 3.32·10-s − 0.0169·11-s + 1.18·12-s − 0.0336·13-s − 1.09·15-s + 1.17·16-s − 1.57·17-s + 0.582·18-s + 0.687·19-s − 3.91·20-s − 0.0296·22-s − 1.55·23-s + 1.06·24-s + 2.62·25-s − 0.0588·26-s + 0.192·27-s + 1.26·29-s − 1.92·30-s − 1.05·31-s + 0.204·32-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 \( 1 - 2.47T + 2T^{2} \)
5 \( 1 + 4.25T + 5T^{2} \)
11 \( 1 + 0.0562T + 11T^{2} \)
13 \( 1 + 0.121T + 13T^{2} \)
17 \( 1 + 6.50T + 17T^{2} \)
19 \( 1 - 2.99T + 19T^{2} \)
23 \( 1 + 7.44T + 23T^{2} \)
29 \( 1 - 6.83T + 29T^{2} \)
31 \( 1 + 5.89T + 31T^{2} \)
37 \( 1 + 3.56T + 37T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 + 12.7T + 47T^{2} \)
53 \( 1 + 2.94T + 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 + 9.07T + 61T^{2} \)
67 \( 1 - 2.83T + 67T^{2} \)
71 \( 1 + 2.41T + 71T^{2} \)
73 \( 1 - 7.14T + 73T^{2} \)
79 \( 1 - 8.01T + 79T^{2} \)
83 \( 1 + 3.64T + 83T^{2} \)
89 \( 1 + 7.76T + 89T^{2} \)
97 \( 1 + 7.07T + 97T^{2} \)
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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

−7.58432672068210784705089859236, −6.82462577901755708623652344315, −6.45254064571558429650439589505, −5.17124289381774372384946570373, −4.64140564290484544549130822766, −3.99005210879597378911181642375, −3.50548737932902155064184239363, −2.82626142305642693788392865416, −1.80645924349365962789650125494, 0, 1.80645924349365962789650125494, 2.82626142305642693788392865416, 3.50548737932902155064184239363, 3.99005210879597378911181642375, 4.64140564290484544549130822766, 5.17124289381774372384946570373, 6.45254064571558429650439589505, 6.82462577901755708623652344315, 7.58432672068210784705089859236

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