Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 9.83e5·2-s − 1.16e9·3-s + 4.17e11·4-s + 1.57e13·5-s + 1.14e15·6-s + 5.39e15·7-s + 1.30e17·8-s + 1.35e18·9-s − 1.55e19·10-s − 1.89e20·11-s − 4.85e20·12-s − 1.17e21·13-s − 5.30e21·14-s − 1.83e22·15-s − 3.57e23·16-s − 1.39e24·17-s − 1.32e24·18-s − 2.79e24·19-s + 6.59e24·20-s − 6.26e24·21-s + 1.86e26·22-s + 5.34e26·23-s − 1.51e26·24-s − 1.56e27·25-s + 1.15e27·26-s − 1.57e27·27-s + 2.25e27·28-s + ⋯
L(s)  = 1  − 1.32·2-s − 0.577·3-s + 0.759·4-s + 0.370·5-s + 0.765·6-s + 0.178·7-s + 0.319·8-s + 0.333·9-s − 0.491·10-s − 0.933·11-s − 0.438·12-s − 0.222·13-s − 0.237·14-s − 0.213·15-s − 1.18·16-s − 1.41·17-s − 0.442·18-s − 0.323·19-s + 0.281·20-s − 0.103·21-s + 1.23·22-s + 1.49·23-s − 0.184·24-s − 0.862·25-s + 0.294·26-s − 0.192·27-s + 0.135·28-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

\(L(20)\) \(\approx\) \(0.5853313267\)
\(L(\frac12)\) \(\approx\) \(0.5853313267\)
\(L(\frac{41}{2})\) 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 \( 1 + 9.83e5T + 5.49e11T^{2} \)
5 \( 1 - 1.57e13T + 1.81e27T^{2} \)
7 \( 1 - 5.39e15T + 9.09e32T^{2} \)
11 \( 1 + 1.89e20T + 4.11e40T^{2} \)
13 \( 1 + 1.17e21T + 2.77e43T^{2} \)
17 \( 1 + 1.39e24T + 9.71e47T^{2} \)
19 \( 1 + 2.79e24T + 7.43e49T^{2} \)
23 \( 1 - 5.34e26T + 1.28e53T^{2} \)
29 \( 1 - 2.12e28T + 1.08e57T^{2} \)
31 \( 1 - 1.34e29T + 1.45e58T^{2} \)
37 \( 1 + 4.34e30T + 1.44e61T^{2} \)
41 \( 1 - 3.17e31T + 7.91e62T^{2} \)
43 \( 1 + 8.86e31T + 5.07e63T^{2} \)
47 \( 1 + 3.38e32T + 1.62e65T^{2} \)
53 \( 1 - 7.04e33T + 1.76e67T^{2} \)
59 \( 1 - 1.96e34T + 1.15e69T^{2} \)
61 \( 1 - 6.79e34T + 4.24e69T^{2} \)
67 \( 1 - 4.10e35T + 1.64e71T^{2} \)
71 \( 1 - 8.69e35T + 1.58e72T^{2} \)
73 \( 1 - 3.39e36T + 4.67e72T^{2} \)
79 \( 1 + 1.47e37T + 1.01e74T^{2} \)
83 \( 1 - 1.98e37T + 6.98e74T^{2} \)
89 \( 1 + 1.95e37T + 1.06e76T^{2} \)
97 \( 1 - 3.46e38T + 3.04e77T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.21091750328857127986392209997, −15.68446044325042690276775077987, −13.27865707835231704204784554121, −11.15666885629649966618385242900, −9.980572378766690460606941198006, −8.451738440920814554881711828451, −6.84169261746180358147463070705, −4.87852124076909544237890976039, −2.15129261232880695177346048504, −0.59979934643266272629309286477, 0.59979934643266272629309286477, 2.15129261232880695177346048504, 4.87852124076909544237890976039, 6.84169261746180358147463070705, 8.451738440920814554881711828451, 9.980572378766690460606941198006, 11.15666885629649966618385242900, 13.27865707835231704204784554121, 15.68446044325042690276775077987, 17.21091750328857127986392209997

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