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  − 0.639·2-s + 3-s − 1.59·4-s − 1.53·5-s − 0.639·6-s − 2.74·7-s + 2.29·8-s + 9-s + 0.979·10-s + 4.38·11-s − 1.59·12-s + 2.19·13-s + 1.75·14-s − 1.53·15-s + 1.71·16-s + 17-s − 0.639·18-s + 4.55·19-s + 2.43·20-s − 2.74·21-s − 2.80·22-s − 7.35·23-s + 2.29·24-s − 2.65·25-s − 1.40·26-s + 27-s + 4.37·28-s + ⋯
L(s)  = 1  − 0.452·2-s + 0.577·3-s − 0.795·4-s − 0.684·5-s − 0.260·6-s − 1.03·7-s + 0.811·8-s + 0.333·9-s + 0.309·10-s + 1.32·11-s − 0.459·12-s + 0.609·13-s + 0.469·14-s − 0.395·15-s + 0.428·16-s + 0.242·17-s − 0.150·18-s + 1.04·19-s + 0.544·20-s − 0.599·21-s − 0.597·22-s − 1.53·23-s + 0.468·24-s − 0.530·25-s − 0.275·26-s + 0.192·27-s + 0.826·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.116384158$
$L(\frac12)$  $\approx$  $1.116384158$
$L(\frac{3}{2})$   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 \( 1 + 0.639T + 2T^{2} \)
5 \( 1 + 1.53T + 5T^{2} \)
7 \( 1 + 2.74T + 7T^{2} \)
11 \( 1 - 4.38T + 11T^{2} \)
13 \( 1 - 2.19T + 13T^{2} \)
19 \( 1 - 4.55T + 19T^{2} \)
23 \( 1 + 7.35T + 23T^{2} \)
29 \( 1 + 1.31T + 29T^{2} \)
31 \( 1 + 5.83T + 31T^{2} \)
37 \( 1 - 4.21T + 37T^{2} \)
41 \( 1 - 3.78T + 41T^{2} \)
43 \( 1 - 3.27T + 43T^{2} \)
47 \( 1 + 5.56T + 47T^{2} \)
53 \( 1 + 1.40T + 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 + 2.49T + 61T^{2} \)
67 \( 1 + 0.208T + 67T^{2} \)
71 \( 1 - 5.71T + 71T^{2} \)
73 \( 1 + 1.55T + 73T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 - 8.42T + 89T^{2} \)
97 \( 1 - 1.90T + 97T^{2} \)
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

−8.502902120887404357378739681179, −7.83022562559898076391794642088, −7.24157413076469581884512293417, −6.30576234950497515558383277566, −5.56045121568908251922442163259, −4.27016767108030607769463051827, −3.81518138587009413884597881522, −3.27673310411858420999476051209, −1.77154837500844076236560625079, −0.65075938329851166663158904462, 0.65075938329851166663158904462, 1.77154837500844076236560625079, 3.27673310411858420999476051209, 3.81518138587009413884597881522, 4.27016767108030607769463051827, 5.56045121568908251922442163259, 6.30576234950497515558383277566, 7.24157413076469581884512293417, 7.83022562559898076391794642088, 8.502902120887404357378739681179

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