Properties

Label 2-6027-1.1-c1-0-142
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.50·2-s + 3-s + 0.272·4-s + 1.10·5-s + 1.50·6-s − 2.60·8-s + 9-s + 1.67·10-s + 3.74·11-s + 0.272·12-s + 4.04·13-s + 1.10·15-s − 4.47·16-s − 1.28·17-s + 1.50·18-s + 2.77·19-s + 0.302·20-s + 5.63·22-s + 8.14·23-s − 2.60·24-s − 3.77·25-s + 6.10·26-s + 27-s − 1.13·29-s + 1.67·30-s + 3.06·31-s − 1.53·32-s + ⋯
L(s)  = 1  + 1.06·2-s + 0.577·3-s + 0.136·4-s + 0.495·5-s + 0.615·6-s − 0.920·8-s + 0.333·9-s + 0.528·10-s + 1.12·11-s + 0.0786·12-s + 1.12·13-s + 0.286·15-s − 1.11·16-s − 0.311·17-s + 0.355·18-s + 0.636·19-s + 0.0675·20-s + 1.20·22-s + 1.69·23-s − 0.531·24-s − 0.754·25-s + 1.19·26-s + 0.192·27-s − 0.210·29-s + 0.305·30-s + 0.550·31-s − 0.270·32-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: \(0\)
Selberg data: \((2,\ 6027,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.960581829\)
\(L(\frac12)\) \(\approx\) \(4.960581829\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.50T + 2T^{2} \)
5 \( 1 - 1.10T + 5T^{2} \)
11 \( 1 - 3.74T + 11T^{2} \)
13 \( 1 - 4.04T + 13T^{2} \)
17 \( 1 + 1.28T + 17T^{2} \)
19 \( 1 - 2.77T + 19T^{2} \)
23 \( 1 - 8.14T + 23T^{2} \)
29 \( 1 + 1.13T + 29T^{2} \)
31 \( 1 - 3.06T + 31T^{2} \)
37 \( 1 + 8.99T + 37T^{2} \)
43 \( 1 - 5.07T + 43T^{2} \)
47 \( 1 + 1.77T + 47T^{2} \)
53 \( 1 + 1.43T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 - 6.29T + 61T^{2} \)
67 \( 1 - 1.53T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 7.32T + 73T^{2} \)
79 \( 1 - 3.06T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 0.862T + 89T^{2} \)
97 \( 1 + 2.95T + 97T^{2} \)
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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

−8.196576188555642333013081976575, −7.09693591158634589835957737457, −6.53120268043903066437234111082, −5.87045995859232305746320398444, −5.12897396953256601503529642622, −4.37270744256177724989735043266, −3.58299192141551857421531208812, −3.16958807003417513503249044151, −2.04053961947627753416557312909, −1.03673219208648237607034090057, 1.03673219208648237607034090057, 2.04053961947627753416557312909, 3.16958807003417513503249044151, 3.58299192141551857421531208812, 4.37270744256177724989735043266, 5.12897396953256601503529642622, 5.87045995859232305746320398444, 6.53120268043903066437234111082, 7.09693591158634589835957737457, 8.196576188555642333013081976575

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