Properties

Degree 2
Conductor 3
Sign $-1$
Motivic weight 37
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 6.79e5·2-s − 3.87e8·3-s + 3.24e11·4-s − 1.35e13·5-s + 2.63e14·6-s − 4.10e15·7-s − 1.26e17·8-s + 1.50e17·9-s + 9.19e18·10-s + 1.15e19·11-s − 1.25e20·12-s + 4.10e20·13-s + 2.78e21·14-s + 5.24e21·15-s + 4.16e22·16-s − 2.52e22·17-s − 1.01e23·18-s + 8.17e23·19-s − 4.38e24·20-s + 1.59e24·21-s − 7.82e24·22-s − 7.67e24·23-s + 4.91e25·24-s + 1.10e26·25-s − 2.79e26·26-s − 5.81e25·27-s − 1.33e27·28-s + ⋯
L(s)  = 1  − 1.83·2-s − 0.577·3-s + 2.35·4-s − 1.58·5-s + 1.05·6-s − 0.952·7-s − 2.49·8-s + 0.333·9-s + 2.90·10-s + 0.624·11-s − 1.36·12-s + 1.01·13-s + 1.74·14-s + 0.915·15-s + 2.20·16-s − 0.435·17-s − 0.610·18-s + 1.80·19-s − 3.74·20-s + 0.550·21-s − 1.14·22-s − 0.493·23-s + 1.43·24-s + 1.51·25-s − 1.85·26-s − 0.192·27-s − 2.24·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(38-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(37\)
character  :  $\chi_{3} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 3,\ (\ :37/2),\ -1)$
$L(19)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{39}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + 3.87e8T \)
good2 \( 1 + 6.79e5T + 1.37e11T^{2} \)
5 \( 1 + 1.35e13T + 7.27e25T^{2} \)
7 \( 1 + 4.10e15T + 1.85e31T^{2} \)
11 \( 1 - 1.15e19T + 3.40e38T^{2} \)
13 \( 1 - 4.10e20T + 1.64e41T^{2} \)
17 \( 1 + 2.52e22T + 3.36e45T^{2} \)
19 \( 1 - 8.17e23T + 2.06e47T^{2} \)
23 \( 1 + 7.67e24T + 2.42e50T^{2} \)
29 \( 1 - 7.52e26T + 1.28e54T^{2} \)
31 \( 1 + 3.41e27T + 1.51e55T^{2} \)
37 \( 1 + 4.82e28T + 1.05e58T^{2} \)
41 \( 1 - 5.14e29T + 4.70e59T^{2} \)
43 \( 1 - 1.56e30T + 2.74e60T^{2} \)
47 \( 1 + 1.18e31T + 7.37e61T^{2} \)
53 \( 1 + 4.07e31T + 6.28e63T^{2} \)
59 \( 1 - 6.63e31T + 3.32e65T^{2} \)
61 \( 1 + 1.38e33T + 1.14e66T^{2} \)
67 \( 1 - 3.53e33T + 3.67e67T^{2} \)
71 \( 1 + 6.34e33T + 3.13e68T^{2} \)
73 \( 1 - 4.38e34T + 8.76e68T^{2} \)
79 \( 1 - 2.43e34T + 1.63e70T^{2} \)
83 \( 1 - 4.41e35T + 1.01e71T^{2} \)
89 \( 1 - 1.79e36T + 1.34e72T^{2} \)
97 \( 1 + 4.46e36T + 3.24e73T^{2} \)
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

−16.25800865321607332381384257567, −15.81092160092555217495271714984, −12.00673509328756648229595396047, −10.99044783344975883596007816580, −9.313913502798474368289953460466, −7.79545021807776993610418432108, −6.55921782783709352127265946708, −3.46420618760276415991557687374, −1.03782586238048776179657920460, 0, 1.03782586238048776179657920460, 3.46420618760276415991557687374, 6.55921782783709352127265946708, 7.79545021807776993610418432108, 9.313913502798474368289953460466, 10.99044783344975883596007816580, 12.00673509328756648229595396047, 15.81092160092555217495271714984, 16.25800865321607332381384257567

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