Properties

Degree $2$
Conductor $3$
Sign $1$
Motivic weight $39$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.64e5·2-s − 1.16e9·3-s − 4.79e11·4-s − 6.30e13·5-s − 3.06e14·6-s − 4.59e16·7-s − 2.71e17·8-s + 1.35e18·9-s − 1.66e19·10-s − 1.42e20·11-s + 5.57e20·12-s + 2.23e20·13-s − 1.21e22·14-s + 7.32e22·15-s + 1.92e23·16-s + 3.05e23·17-s + 3.56e23·18-s − 1.47e25·19-s + 3.02e25·20-s + 5.33e25·21-s − 3.77e25·22-s − 5.75e26·23-s + 3.16e26·24-s + 2.15e27·25-s + 5.90e25·26-s − 1.57e27·27-s + 2.20e28·28-s + ⋯
L(s)  = 1  + 0.356·2-s − 0.577·3-s − 0.873·4-s − 1.47·5-s − 0.205·6-s − 1.52·7-s − 0.667·8-s + 0.333·9-s − 0.526·10-s − 0.704·11-s + 0.504·12-s + 0.0424·13-s − 0.542·14-s + 0.853·15-s + 0.635·16-s + 0.309·17-s + 0.118·18-s − 1.70·19-s + 1.29·20-s + 0.878·21-s − 0.250·22-s − 1.60·23-s + 0.385·24-s + 1.18·25-s + 0.0151·26-s − 0.192·27-s + 1.32·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Motivic weight: \(39\)
Character: $\chi_{3} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :39/2),\ 1)\)

Particular Values

\(L(20)\) \(\approx\) \(0.007480562113\)
\(L(\frac12)\) \(\approx\) \(0.007480562113\)
\(L(\frac{41}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.16e9T \)
good2 \( 1 - 2.64e5T + 5.49e11T^{2} \)
5 \( 1 + 6.30e13T + 1.81e27T^{2} \)
7 \( 1 + 4.59e16T + 9.09e32T^{2} \)
11 \( 1 + 1.42e20T + 4.11e40T^{2} \)
13 \( 1 - 2.23e20T + 2.77e43T^{2} \)
17 \( 1 - 3.05e23T + 9.71e47T^{2} \)
19 \( 1 + 1.47e25T + 7.43e49T^{2} \)
23 \( 1 + 5.75e26T + 1.28e53T^{2} \)
29 \( 1 - 1.59e28T + 1.08e57T^{2} \)
31 \( 1 - 2.90e28T + 1.45e58T^{2} \)
37 \( 1 - 4.49e30T + 1.44e61T^{2} \)
41 \( 1 + 3.98e31T + 7.91e62T^{2} \)
43 \( 1 + 1.13e32T + 5.07e63T^{2} \)
47 \( 1 + 4.87e32T + 1.62e65T^{2} \)
53 \( 1 + 5.94e32T + 1.76e67T^{2} \)
59 \( 1 + 2.61e34T + 1.15e69T^{2} \)
61 \( 1 + 4.20e34T + 4.24e69T^{2} \)
67 \( 1 + 4.97e35T + 1.64e71T^{2} \)
71 \( 1 - 8.30e35T + 1.58e72T^{2} \)
73 \( 1 + 5.71e35T + 4.67e72T^{2} \)
79 \( 1 + 1.34e37T + 1.01e74T^{2} \)
83 \( 1 - 2.31e37T + 6.98e74T^{2} \)
89 \( 1 - 5.53e37T + 1.06e76T^{2} \)
97 \( 1 + 5.58e38T + 3.04e77T^{2} \)
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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

−16.49005129973931558504073827616, −15.27553986095562255720854384855, −13.10662426979785944794974144451, −12.08736538840444288721523391180, −10.11272813802674521204998487875, −8.202488053188601532799446059622, −6.29219727325417041678329990124, −4.46148901211808124548251909046, −3.35303034384620127315404256382, −0.05062235783673003003673606429, 0.05062235783673003003673606429, 3.35303034384620127315404256382, 4.46148901211808124548251909046, 6.29219727325417041678329990124, 8.202488053188601532799446059622, 10.11272813802674521204998487875, 12.08736538840444288721523391180, 13.10662426979785944794974144451, 15.27553986095562255720854384855, 16.49005129973931558504073827616

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