Properties

Label 2-1-1.1-c27-0-0
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $4.61855$
Root an. cond. $2.14908$
Motivic weight $27$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.87e4·2-s − 3.44e6·3-s + 2.15e8·4-s + 5.76e8·5-s + 6.43e10·6-s − 1.96e11·7-s − 1.53e12·8-s + 4.21e12·9-s − 1.07e13·10-s + 2.06e14·11-s − 7.43e14·12-s − 1.66e14·13-s + 3.68e15·14-s − 1.98e15·15-s − 3.55e14·16-s − 5.47e15·17-s − 7.89e16·18-s + 1.61e17·19-s + 1.24e17·20-s + 6.77e17·21-s − 3.86e18·22-s + 2.80e18·23-s + 5.26e18·24-s − 7.11e18·25-s + 3.11e18·26-s + 1.17e19·27-s − 4.25e19·28-s + ⋯
L(s)  = 1  − 1.61·2-s − 1.24·3-s + 1.60·4-s + 0.211·5-s + 2.01·6-s − 0.768·7-s − 0.984·8-s + 0.552·9-s − 0.341·10-s + 1.80·11-s − 2.00·12-s − 0.152·13-s + 1.24·14-s − 0.263·15-s − 0.0197·16-s − 0.133·17-s − 0.893·18-s + 0.880·19-s + 0.339·20-s + 0.957·21-s − 2.91·22-s + 1.15·23-s + 1.22·24-s − 0.955·25-s + 0.246·26-s + 0.557·27-s − 1.23·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(28-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+27/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(4.61855\)
Root analytic conductor: \(2.14908\)
Motivic weight: \(27\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(0.4313634313\)
\(L(\frac12)\) \(\approx\) \(0.4313634313\)
\(L(\frac{29}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.87e4T + 1.34e8T^{2} \)
3 \( 1 + 3.44e6T + 7.62e12T^{2} \)
5 \( 1 - 5.76e8T + 7.45e18T^{2} \)
7 \( 1 + 1.96e11T + 6.57e22T^{2} \)
11 \( 1 - 2.06e14T + 1.31e28T^{2} \)
13 \( 1 + 1.66e14T + 1.19e30T^{2} \)
17 \( 1 + 5.47e15T + 1.66e33T^{2} \)
19 \( 1 - 1.61e17T + 3.36e34T^{2} \)
23 \( 1 - 2.80e18T + 5.84e36T^{2} \)
29 \( 1 + 2.99e18T + 3.05e39T^{2} \)
31 \( 1 - 9.09e19T + 1.84e40T^{2} \)
37 \( 1 - 1.50e21T + 2.19e42T^{2} \)
41 \( 1 - 5.47e21T + 3.50e43T^{2} \)
43 \( 1 + 6.81e21T + 1.26e44T^{2} \)
47 \( 1 + 9.41e21T + 1.40e45T^{2} \)
53 \( 1 + 5.35e22T + 3.59e46T^{2} \)
59 \( 1 - 9.16e23T + 6.50e47T^{2} \)
61 \( 1 - 1.47e24T + 1.59e48T^{2} \)
67 \( 1 + 3.21e24T + 2.01e49T^{2} \)
71 \( 1 - 3.38e24T + 9.63e49T^{2} \)
73 \( 1 - 1.32e25T + 2.04e50T^{2} \)
79 \( 1 - 4.66e25T + 1.72e51T^{2} \)
83 \( 1 - 7.14e25T + 6.53e51T^{2} \)
89 \( 1 + 1.40e26T + 4.30e52T^{2} \)
97 \( 1 - 3.77e26T + 4.39e53T^{2} \)
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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

−27.10563959812033108832731117675, −25.11750685583981067778108451479, −22.50188545029103663195437511378, −19.50265774061765824385221797449, −17.59210966024527609444600839401, −16.50582946123084970950610092127, −11.52175848608226414749693302062, −9.537244640310907359419777634061, −6.56510505470169432955925772781, −0.877191859570262290319324466835, 0.877191859570262290319324466835, 6.56510505470169432955925772781, 9.537244640310907359419777634061, 11.52175848608226414749693302062, 16.50582946123084970950610092127, 17.59210966024527609444600839401, 19.50265774061765824385221797449, 22.50188545029103663195437511378, 25.11750685583981067778108451479, 27.10563959812033108832731117675

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