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  + 1.03e16·2-s − 2.77e24·3-s + 6.60e31·4-s − 1.38e36·5-s − 2.86e40·6-s − 3.04e43·7-s + 2.63e47·8-s − 1.17e50·9-s − 1.43e52·10-s + 4.98e54·11-s − 1.83e56·12-s + 2.40e58·13-s − 3.14e59·14-s + 3.85e60·15-s + 4.01e61·16-s − 3.48e64·17-s − 1.21e66·18-s − 1.66e67·19-s − 9.17e67·20-s + 8.44e67·21-s + 5.14e70·22-s − 2.46e70·23-s − 7.30e71·24-s − 2.27e73·25-s + 2.48e74·26-s + 6.73e74·27-s − 2.01e75·28-s + ⋯
L(s)  = 1  + 1.62·2-s − 0.247·3-s + 1.62·4-s − 0.279·5-s − 0.401·6-s − 0.130·7-s + 1.01·8-s − 0.938·9-s − 0.453·10-s + 1.05·11-s − 0.403·12-s + 0.791·13-s − 0.211·14-s + 0.0693·15-s + 0.0244·16-s − 0.879·17-s − 1.52·18-s − 1.22·19-s − 0.455·20-s + 0.0323·21-s + 1.71·22-s − 0.0795·23-s − 0.252·24-s − 0.921·25-s + 1.28·26-s + 0.480·27-s − 0.212·28-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 \( 1 - 1.03e16T + 4.05e31T^{2} \)
3 \( 1 + 2.77e24T + 1.25e50T^{2} \)
5 \( 1 + 1.38e36T + 2.46e73T^{2} \)
7 \( 1 + 3.04e43T + 5.43e88T^{2} \)
11 \( 1 - 4.98e54T + 2.21e109T^{2} \)
13 \( 1 - 2.40e58T + 9.20e116T^{2} \)
17 \( 1 + 3.48e64T + 1.57e129T^{2} \)
19 \( 1 + 1.66e67T + 1.85e134T^{2} \)
23 \( 1 + 2.46e70T + 9.58e142T^{2} \)
29 \( 1 + 7.76e76T + 3.56e153T^{2} \)
31 \( 1 - 1.96e78T + 3.91e156T^{2} \)
37 \( 1 + 3.69e82T + 4.58e164T^{2} \)
41 \( 1 + 4.85e84T + 2.19e169T^{2} \)
43 \( 1 - 9.79e85T + 3.26e171T^{2} \)
47 \( 1 + 3.79e87T + 3.71e175T^{2} \)
53 \( 1 - 1.90e90T + 1.11e181T^{2} \)
59 \( 1 + 2.84e92T + 8.69e185T^{2} \)
61 \( 1 - 3.64e93T + 2.88e187T^{2} \)
67 \( 1 + 4.96e95T + 5.46e191T^{2} \)
71 \( 1 - 1.33e97T + 2.41e194T^{2} \)
73 \( 1 + 8.83e97T + 4.45e195T^{2} \)
79 \( 1 + 1.15e99T + 1.78e199T^{2} \)
83 \( 1 + 5.00e100T + 3.18e201T^{2} \)
89 \( 1 - 6.61e101T + 4.85e204T^{2} \)
97 \( 1 - 1.82e104T + 4.08e208T^{2} \)
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

−13.34937577677660313719553505316, −11.96512864781676150786227663542, −11.07369036943310906136249180993, −8.732243789616008284186091035356, −6.65566115076316911901412442783, −5.79795191873918231946372269808, −4.34256204433287742239635233014, −3.42395226448728700774421044460, −1.96962397760348390164621000186, 0, 1.96962397760348390164621000186, 3.42395226448728700774421044460, 4.34256204433287742239635233014, 5.79795191873918231946372269808, 6.65566115076316911901412442783, 8.732243789616008284186091035356, 11.07369036943310906136249180993, 11.96512864781676150786227663542, 13.34937577677660313719553505316

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