Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 105
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.15e16·2-s + 2.00e25·3-s + 9.24e31·4-s − 4.02e36·5-s − 2.31e41·6-s − 1.49e43·7-s − 5.98e47·8-s + 2.76e50·9-s + 4.64e52·10-s − 4.19e54·11-s + 1.85e57·12-s + 4.35e58·13-s + 1.72e59·14-s − 8.06e61·15-s + 3.15e63·16-s − 5.45e64·17-s − 3.18e66·18-s + 5.82e66·19-s − 3.72e68·20-s − 2.99e68·21-s + 4.83e70·22-s − 5.38e70·23-s − 1.19e73·24-s − 8.43e72·25-s − 5.02e74·26-s + 3.02e75·27-s − 1.38e75·28-s + ⋯
L(s)  = 1  − 1.81·2-s + 1.79·3-s + 2.27·4-s − 0.810·5-s − 3.24·6-s − 0.0640·7-s − 2.31·8-s + 2.20·9-s + 1.46·10-s − 0.889·11-s + 4.08·12-s + 1.43·13-s + 0.116·14-s − 1.45·15-s + 1.91·16-s − 1.37·17-s − 3.99·18-s + 0.427·19-s − 1.84·20-s − 0.114·21-s + 1.61·22-s − 0.173·23-s − 4.14·24-s − 0.342·25-s − 2.60·26-s + 2.15·27-s − 0.146·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(106-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+52.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(105\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :105/2),\ -1)\)
\(L(53)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{107}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 + 1.15e16T + 4.05e31T^{2} \)
3 \( 1 - 2.00e25T + 1.25e50T^{2} \)
5 \( 1 + 4.02e36T + 2.46e73T^{2} \)
7 \( 1 + 1.49e43T + 5.43e88T^{2} \)
11 \( 1 + 4.19e54T + 2.21e109T^{2} \)
13 \( 1 - 4.35e58T + 9.20e116T^{2} \)
17 \( 1 + 5.45e64T + 1.57e129T^{2} \)
19 \( 1 - 5.82e66T + 1.85e134T^{2} \)
23 \( 1 + 5.38e70T + 9.58e142T^{2} \)
29 \( 1 + 5.63e76T + 3.56e153T^{2} \)
31 \( 1 - 1.44e78T + 3.91e156T^{2} \)
37 \( 1 + 2.11e81T + 4.58e164T^{2} \)
41 \( 1 + 5.42e84T + 2.19e169T^{2} \)
43 \( 1 + 3.33e85T + 3.26e171T^{2} \)
47 \( 1 - 2.06e87T + 3.71e175T^{2} \)
53 \( 1 + 1.52e90T + 1.11e181T^{2} \)
59 \( 1 - 1.70e93T + 8.69e185T^{2} \)
61 \( 1 - 2.17e93T + 2.88e187T^{2} \)
67 \( 1 - 8.39e94T + 5.46e191T^{2} \)
71 \( 1 - 1.45e97T + 2.41e194T^{2} \)
73 \( 1 - 3.40e97T + 4.45e195T^{2} \)
79 \( 1 + 3.48e99T + 1.78e199T^{2} \)
83 \( 1 + 4.88e100T + 3.18e201T^{2} \)
89 \( 1 + 3.37e102T + 4.85e204T^{2} \)
97 \( 1 + 3.41e104T + 4.08e208T^{2} \)
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

−13.37162218720617542277827007968, −11.17876087964746311821930897144, −9.791135979916819444995487467127, −8.558006006440844100959177091895, −8.103607090278107896128053663514, −6.93021865871460886614165559316, −3.74372658021979788922219480439, −2.55300900994618047188007980970, −1.48051452513117376829966190964, 0, 1.48051452513117376829966190964, 2.55300900994618047188007980970, 3.74372658021979788922219480439, 6.93021865871460886614165559316, 8.103607090278107896128053663514, 8.558006006440844100959177091895, 9.791135979916819444995487467127, 11.17876087964746311821930897144, 13.37162218720617542277827007968

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