Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 101
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.29e15·2-s − 2.26e24·3-s − 8.55e29·4-s − 1.52e35·5-s − 2.93e39·6-s − 2.30e42·7-s − 4.39e45·8-s + 3.57e48·9-s − 1.98e50·10-s + 7.37e52·11-s + 1.93e54·12-s + 1.08e56·13-s − 2.98e57·14-s + 3.46e59·15-s − 3.52e60·16-s − 7.80e61·17-s + 4.63e63·18-s − 2.75e64·19-s + 1.30e65·20-s + 5.21e66·21-s + 9.56e67·22-s + 8.02e68·23-s + 9.94e69·24-s − 1.60e70·25-s + 1.40e71·26-s − 4.59e72·27-s + 1.97e72·28-s + ⋯
L(s)  = 1  + 0.813·2-s − 1.82·3-s − 0.337·4-s − 0.770·5-s − 1.48·6-s − 0.483·7-s − 1.08·8-s + 2.31·9-s − 0.626·10-s + 1.89·11-s + 0.614·12-s + 0.602·13-s − 0.393·14-s + 1.40·15-s − 0.548·16-s − 0.568·17-s + 1.88·18-s − 0.729·19-s + 0.260·20-s + 0.880·21-s + 1.54·22-s + 1.37·23-s + 1.98·24-s − 0.406·25-s + 0.490·26-s − 2.38·27-s + 0.163·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(102-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+50.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(101\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :101/2),\ -1)\)
\(L(51)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{103}{2})\)   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 \( 1 - 1.29e15T + 2.53e30T^{2} \)
3 \( 1 + 2.26e24T + 1.54e48T^{2} \)
5 \( 1 + 1.52e35T + 3.94e70T^{2} \)
7 \( 1 + 2.30e42T + 2.26e85T^{2} \)
11 \( 1 - 7.37e52T + 1.51e105T^{2} \)
13 \( 1 - 1.08e56T + 3.22e112T^{2} \)
17 \( 1 + 7.80e61T + 1.88e124T^{2} \)
19 \( 1 + 2.75e64T + 1.42e129T^{2} \)
23 \( 1 - 8.02e68T + 3.42e137T^{2} \)
29 \( 1 + 4.17e73T + 5.03e147T^{2} \)
31 \( 1 - 1.32e74T + 4.24e150T^{2} \)
37 \( 1 - 7.61e78T + 2.44e158T^{2} \)
41 \( 1 - 4.16e81T + 7.78e162T^{2} \)
43 \( 1 + 6.23e81T + 9.55e164T^{2} \)
47 \( 1 + 9.39e83T + 7.61e168T^{2} \)
53 \( 1 - 1.74e87T + 1.41e174T^{2} \)
59 \( 1 - 2.25e89T + 7.17e178T^{2} \)
61 \( 1 + 2.74e90T + 2.08e180T^{2} \)
67 \( 1 - 8.52e91T + 2.71e184T^{2} \)
71 \( 1 + 3.72e93T + 9.48e186T^{2} \)
73 \( 1 + 2.87e93T + 1.56e188T^{2} \)
79 \( 1 - 4.49e95T + 4.57e191T^{2} \)
83 \( 1 - 1.36e96T + 6.71e193T^{2} \)
89 \( 1 - 2.37e98T + 7.73e196T^{2} \)
97 \( 1 + 7.96e99T + 4.61e200T^{2} \)
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

−13.07671410039418899376880312522, −11.99544613062561610326758848976, −11.14342480777275099157871844788, −9.214437841642332248701727031610, −6.75363602026425118950790788001, −5.94131937938569002000718710016, −4.49509318553229487084691255763, −3.78987939913398560764337013948, −1.04157930115495124271021096389, 0, 1.04157930115495124271021096389, 3.78987939913398560764337013948, 4.49509318553229487084691255763, 5.94131937938569002000718710016, 6.75363602026425118950790788001, 9.214437841642332248701727031610, 11.14342480777275099157871844788, 11.99544613062561610326758848976, 13.07671410039418899376880312522

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