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  − 1.26·2-s + 3-s − 0.394·4-s + 3.60·5-s − 1.26·6-s − 2.46·7-s + 3.03·8-s + 9-s − 4.56·10-s + 4.32·11-s − 0.394·12-s + 6.87·13-s + 3.11·14-s + 3.60·15-s − 3.05·16-s + 17-s − 1.26·18-s + 1.02·19-s − 1.42·20-s − 2.46·21-s − 5.48·22-s + 3.18·23-s + 3.03·24-s + 8.00·25-s − 8.71·26-s + 27-s + 0.972·28-s + ⋯
L(s)  = 1  − 0.895·2-s + 0.577·3-s − 0.197·4-s + 1.61·5-s − 0.517·6-s − 0.930·7-s + 1.07·8-s + 0.333·9-s − 1.44·10-s + 1.30·11-s − 0.114·12-s + 1.90·13-s + 0.833·14-s + 0.931·15-s − 0.763·16-s + 0.242·17-s − 0.298·18-s + 0.234·19-s − 0.318·20-s − 0.537·21-s − 1.16·22-s + 0.663·23-s + 0.619·24-s + 1.60·25-s − 1.70·26-s + 0.192·27-s + 0.183·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.192140391$
$L(\frac12)$  $\approx$  $2.192140391$
$L(\frac{3}{2})$   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 \( 1 + 1.26T + 2T^{2} \)
5 \( 1 - 3.60T + 5T^{2} \)
7 \( 1 + 2.46T + 7T^{2} \)
11 \( 1 - 4.32T + 11T^{2} \)
13 \( 1 - 6.87T + 13T^{2} \)
19 \( 1 - 1.02T + 19T^{2} \)
23 \( 1 - 3.18T + 23T^{2} \)
29 \( 1 - 2.10T + 29T^{2} \)
31 \( 1 - 11.0T + 31T^{2} \)
37 \( 1 + 3.84T + 37T^{2} \)
41 \( 1 + 3.71T + 41T^{2} \)
43 \( 1 - 5.79T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + 1.67T + 53T^{2} \)
59 \( 1 - 1.63T + 59T^{2} \)
61 \( 1 + 7.36T + 61T^{2} \)
67 \( 1 - 2.03T + 67T^{2} \)
71 \( 1 + 3.93T + 71T^{2} \)
73 \( 1 + 7.02T + 73T^{2} \)
83 \( 1 - 2.00T + 83T^{2} \)
89 \( 1 - 3.75T + 89T^{2} \)
97 \( 1 + 0.409T + 97T^{2} \)
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

−8.655885502466408628910104503288, −8.096792182543997194603873782806, −6.72702975398398657371493330742, −6.52822662287221379026819408055, −5.71153877617535070215385324370, −4.62045732098024090968431219722, −3.66582905058549205386454393653, −2.86954091799567498220942646755, −1.53391396807334448057760682589, −1.13562048764686207624853076675, 1.13562048764686207624853076675, 1.53391396807334448057760682589, 2.86954091799567498220942646755, 3.66582905058549205386454393653, 4.62045732098024090968431219722, 5.71153877617535070215385324370, 6.52822662287221379026819408055, 6.72702975398398657371493330742, 8.096792182543997194603873782806, 8.655885502466408628910104503288

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