Properties

Degree 2
Conductor 3
Sign $1$
Motivic weight 41
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.59e6·2-s + 3.48e9·3-s + 3.59e11·4-s + 3.20e14·5-s − 5.57e15·6-s + 2.35e17·7-s + 2.94e18·8-s + 1.21e19·9-s − 5.13e20·10-s + 4.09e21·11-s + 1.25e21·12-s + 2.26e22·13-s − 3.76e23·14-s + 1.11e24·15-s − 5.49e24·16-s − 2.18e25·17-s − 1.94e25·18-s − 1.65e26·19-s + 1.15e26·20-s + 8.21e26·21-s − 6.54e27·22-s + 1.04e28·23-s + 1.02e28·24-s + 5.75e28·25-s − 3.61e28·26-s + 4.23e28·27-s + 8.45e28·28-s + ⋯
L(s)  = 1  − 1.07·2-s + 0.577·3-s + 0.163·4-s + 1.50·5-s − 0.622·6-s + 1.11·7-s + 0.902·8-s + 0.333·9-s − 1.62·10-s + 1.83·11-s + 0.0942·12-s + 0.330·13-s − 1.20·14-s + 0.868·15-s − 1.13·16-s − 1.30·17-s − 0.359·18-s − 1.00·19-s + 0.245·20-s + 0.643·21-s − 1.97·22-s + 1.27·23-s + 0.521·24-s + 1.26·25-s − 0.356·26-s + 0.192·27-s + 0.182·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(41\)
character  :  $\chi_{3} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3,\ (\ :41/2),\ 1)\)
\(L(21)\)  \(\approx\)  \(2.226565486\)
\(L(\frac12)\)  \(\approx\)  \(2.226565486\)
\(L(\frac{43}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - 3.48e9T \)
good2 \( 1 + 1.59e6T + 2.19e12T^{2} \)
5 \( 1 - 3.20e14T + 4.54e28T^{2} \)
7 \( 1 - 2.35e17T + 4.45e34T^{2} \)
11 \( 1 - 4.09e21T + 4.97e42T^{2} \)
13 \( 1 - 2.26e22T + 4.69e45T^{2} \)
17 \( 1 + 2.18e25T + 2.80e50T^{2} \)
19 \( 1 + 1.65e26T + 2.68e52T^{2} \)
23 \( 1 - 1.04e28T + 6.77e55T^{2} \)
29 \( 1 + 1.48e29T + 9.08e59T^{2} \)
31 \( 1 - 2.32e30T + 1.39e61T^{2} \)
37 \( 1 + 5.37e31T + 1.97e64T^{2} \)
41 \( 1 + 9.27e32T + 1.33e66T^{2} \)
43 \( 1 - 4.47e33T + 9.38e66T^{2} \)
47 \( 1 + 1.02e34T + 3.59e68T^{2} \)
53 \( 1 + 2.32e35T + 4.95e70T^{2} \)
59 \( 1 + 1.00e35T + 4.02e72T^{2} \)
61 \( 1 - 1.56e36T + 1.57e73T^{2} \)
67 \( 1 + 5.00e36T + 7.39e74T^{2} \)
71 \( 1 + 7.95e37T + 7.97e75T^{2} \)
73 \( 1 + 1.56e38T + 2.49e76T^{2} \)
79 \( 1 - 1.20e39T + 6.34e77T^{2} \)
83 \( 1 - 2.45e39T + 4.81e78T^{2} \)
89 \( 1 + 1.06e40T + 8.41e79T^{2} \)
97 \( 1 - 2.08e40T + 2.86e81T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.14737312729709694774644587317, −14.56788147671430719647916049564, −13.47530450590324919211464740544, −10.90466941294547704442078424401, −9.340635324934120044071582964261, −8.631396541114538771867213723687, −6.64583078675594647141102048634, −4.51451479554662000466001581882, −1.97480057373067621755313571870, −1.24455453507659229966356001716, 1.24455453507659229966356001716, 1.97480057373067621755313571870, 4.51451479554662000466001581882, 6.64583078675594647141102048634, 8.631396541114538771867213723687, 9.340635324934120044071582964261, 10.90466941294547704442078424401, 13.47530450590324919211464740544, 14.56788147671430719647916049564, 17.14737312729709694774644587317

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