Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 79 $
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  + 0.0488·2-s + 3-s − 1.99·4-s − 0.188·5-s + 0.0488·6-s + 3.46·7-s − 0.195·8-s + 9-s − 0.00918·10-s + 3.08·11-s − 1.99·12-s − 6.69·13-s + 0.169·14-s − 0.188·15-s + 3.98·16-s + 17-s + 0.0488·18-s − 1.82·19-s + 0.375·20-s + 3.46·21-s + 0.150·22-s + 3.48·23-s − 0.195·24-s − 4.96·25-s − 0.326·26-s + 27-s − 6.92·28-s + ⋯
L(s)  = 1  + 0.0345·2-s + 0.577·3-s − 0.998·4-s − 0.0841·5-s + 0.0199·6-s + 1.31·7-s − 0.0690·8-s + 0.333·9-s − 0.00290·10-s + 0.930·11-s − 0.576·12-s − 1.85·13-s + 0.0452·14-s − 0.0485·15-s + 0.996·16-s + 0.242·17-s + 0.0115·18-s − 0.417·19-s + 0.0840·20-s + 0.756·21-s + 0.0321·22-s + 0.727·23-s − 0.0398·24-s − 0.992·25-s − 0.0640·26-s + 0.192·27-s − 1.30·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4029\)    =    \(3 \cdot 17 \cdot 79\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4029} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.150831605$
$L(\frac12)$  $\approx$  $2.150831605$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;17,\;79\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;79\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
17 \( 1 - T \)
79 \( 1 - T \)
good2 \( 1 - 0.0488T + 2T^{2} \)
5 \( 1 + 0.188T + 5T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 - 3.08T + 11T^{2} \)
13 \( 1 + 6.69T + 13T^{2} \)
19 \( 1 + 1.82T + 19T^{2} \)
23 \( 1 - 3.48T + 23T^{2} \)
29 \( 1 + 4.01T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 - 8.47T + 37T^{2} \)
41 \( 1 - 5.10T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 8.85T + 47T^{2} \)
53 \( 1 + 8.03T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 - 6.94T + 61T^{2} \)
67 \( 1 + 1.50T + 67T^{2} \)
71 \( 1 + 9.99T + 71T^{2} \)
73 \( 1 + 2.67T + 73T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 - 4.05T + 89T^{2} \)
97 \( 1 - 8.47T + 97T^{2} \)
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

−8.403797605328753141945284288901, −7.79136116288816047182649377007, −7.33481258376302523791485575333, −6.16001969278663295536515876364, −5.21029168524085341315456588567, −4.48434497827592718665217265059, −4.17870606083739669426439926545, −2.93710973464948372311885947629, −1.97788850260818201409717284768, −0.847274792310917613042514131527, 0.847274792310917613042514131527, 1.97788850260818201409717284768, 2.93710973464948372311885947629, 4.17870606083739669426439926545, 4.48434497827592718665217265059, 5.21029168524085341315456588567, 6.16001969278663295536515876364, 7.33481258376302523791485575333, 7.79136116288816047182649377007, 8.403797605328753141945284288901

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