Properties

Label 2-6027-1.1-c1-0-229
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  + 2.54·2-s + 3-s + 4.48·4-s + 2.36·5-s + 2.54·6-s + 6.31·8-s + 9-s + 6.01·10-s + 2.84·11-s + 4.48·12-s − 3.22·13-s + 2.36·15-s + 7.11·16-s − 3.62·17-s + 2.54·18-s − 1.65·19-s + 10.5·20-s + 7.23·22-s + 6.36·23-s + 6.31·24-s + 0.577·25-s − 8.21·26-s + 27-s − 2.75·29-s + 6.01·30-s + 7.33·31-s + 5.49·32-s + ⋯
L(s)  = 1  + 1.80·2-s + 0.577·3-s + 2.24·4-s + 1.05·5-s + 1.03·6-s + 2.23·8-s + 0.333·9-s + 1.90·10-s + 0.857·11-s + 1.29·12-s − 0.895·13-s + 0.609·15-s + 1.77·16-s − 0.879·17-s + 0.600·18-s − 0.378·19-s + 2.36·20-s + 1.54·22-s + 1.32·23-s + 1.28·24-s + 0.115·25-s − 1.61·26-s + 0.192·27-s − 0.510·29-s + 1.09·30-s + 1.31·31-s + 0.970·32-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

\(L(1)\) \(\approx\) \(9.913687666\)
\(L(\frac12)\) \(\approx\) \(9.913687666\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.54T + 2T^{2} \)
5 \( 1 - 2.36T + 5T^{2} \)
11 \( 1 - 2.84T + 11T^{2} \)
13 \( 1 + 3.22T + 13T^{2} \)
17 \( 1 + 3.62T + 17T^{2} \)
19 \( 1 + 1.65T + 19T^{2} \)
23 \( 1 - 6.36T + 23T^{2} \)
29 \( 1 + 2.75T + 29T^{2} \)
31 \( 1 - 7.33T + 31T^{2} \)
37 \( 1 - 2.24T + 37T^{2} \)
43 \( 1 - 0.638T + 43T^{2} \)
47 \( 1 - 13.6T + 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 - 3.75T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 5.35T + 67T^{2} \)
71 \( 1 + 7.48T + 71T^{2} \)
73 \( 1 + 2.79T + 73T^{2} \)
79 \( 1 - 7.82T + 79T^{2} \)
83 \( 1 + 3.76T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + 9.84T + 97T^{2} \)
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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

−7.75629969323146459989752981955, −7.03466183977986709150919202496, −6.43624949294037436506821226002, −5.92971395363746601252271454574, −5.00650213249545759276557655768, −4.51328849396837791024970531098, −3.75825777288521191574581544760, −2.75991820994189816264611840850, −2.34634230079349469440253768781, −1.41926032489317530332869323248, 1.41926032489317530332869323248, 2.34634230079349469440253768781, 2.75991820994189816264611840850, 3.75825777288521191574581544760, 4.51328849396837791024970531098, 5.00650213249545759276557655768, 5.92971395363746601252271454574, 6.43624949294037436506821226002, 7.03466183977986709150919202496, 7.75629969323146459989752981955

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