Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 149 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 0.448·5-s − 1.52·7-s − 8-s − 0.448·10-s − 0.412·11-s + 3.36·13-s + 1.52·14-s + 16-s + 3.96·17-s − 5.13·19-s + 0.448·20-s + 0.412·22-s − 2.63·23-s − 4.79·25-s − 3.36·26-s − 1.52·28-s − 2.15·29-s − 5.88·31-s − 32-s − 3.96·34-s − 0.686·35-s + 10.4·37-s + 5.13·38-s − 0.448·40-s − 5.89·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.200·5-s − 0.578·7-s − 0.353·8-s − 0.141·10-s − 0.124·11-s + 0.933·13-s + 0.408·14-s + 0.250·16-s + 0.961·17-s − 1.17·19-s + 0.100·20-s + 0.0878·22-s − 0.549·23-s − 0.959·25-s − 0.659·26-s − 0.289·28-s − 0.399·29-s − 1.05·31-s − 0.176·32-s − 0.680·34-s − 0.116·35-s + 1.72·37-s + 0.832·38-s − 0.0709·40-s − 0.920·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8046\)    =    \(2 \cdot 3^{3} \cdot 149\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8046} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.172255923$
$L(\frac12)$  $\approx$  $1.172255923$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;149\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;149\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
149 \( 1 - T \)
good5 \( 1 - 0.448T + 5T^{2} \)
7 \( 1 + 1.52T + 7T^{2} \)
11 \( 1 + 0.412T + 11T^{2} \)
13 \( 1 - 3.36T + 13T^{2} \)
17 \( 1 - 3.96T + 17T^{2} \)
19 \( 1 + 5.13T + 19T^{2} \)
23 \( 1 + 2.63T + 23T^{2} \)
29 \( 1 + 2.15T + 29T^{2} \)
31 \( 1 + 5.88T + 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 + 5.89T + 41T^{2} \)
43 \( 1 - 1.10T + 43T^{2} \)
47 \( 1 - 5.78T + 47T^{2} \)
53 \( 1 + 1.69T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 15.8T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 - 5.42T + 73T^{2} \)
79 \( 1 - 2.21T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 - 0.410T + 89T^{2} \)
97 \( 1 - 4.08T + 97T^{2} \)
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

−7.960865904007374876264147644915, −7.23763747353153598443852685067, −6.38040381725908857039024478784, −5.99228815736715506511393989859, −5.25943166075589544539360532830, −4.01157719949299304487884685671, −3.54935941665052122806013791530, −2.47901431769468590366752572893, −1.72052007214366446129820805432, −0.59233792884023619034504634310, 0.59233792884023619034504634310, 1.72052007214366446129820805432, 2.47901431769468590366752572893, 3.54935941665052122806013791530, 4.01157719949299304487884685671, 5.25943166075589544539360532830, 5.99228815736715506511393989859, 6.38040381725908857039024478784, 7.23763747353153598443852685067, 7.960865904007374876264147644915

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