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  − 2.65e15·2-s − 2.23e24·3-s + 4.53e30·4-s + 2.85e35·5-s + 5.95e39·6-s + 3.30e41·7-s − 5.31e45·8-s + 3.46e48·9-s − 7.60e50·10-s − 4.96e52·11-s − 1.01e55·12-s + 5.64e55·13-s − 8.78e56·14-s − 6.40e59·15-s + 2.62e60·16-s − 1.94e61·17-s − 9.22e63·18-s + 3.52e64·19-s + 1.29e66·20-s − 7.39e65·21-s + 1.32e68·22-s − 1.63e68·23-s + 1.18e70·24-s + 4.23e70·25-s − 1.50e71·26-s − 4.30e72·27-s + 1.49e72·28-s + ⋯
L(s)  = 1  − 1.66·2-s − 1.80·3-s + 1.78·4-s + 1.43·5-s + 3.00·6-s + 0.0694·7-s − 1.31·8-s + 2.24·9-s − 2.40·10-s − 1.27·11-s − 3.22·12-s + 0.314·13-s − 0.115·14-s − 2.59·15-s + 0.409·16-s − 0.141·17-s − 3.74·18-s + 0.932·19-s + 2.57·20-s − 0.125·21-s + 2.12·22-s − 0.279·23-s + 2.37·24-s + 1.07·25-s − 0.524·26-s − 2.24·27-s + 0.124·28-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 \( 1 + 2.65e15T + 2.53e30T^{2} \)
3 \( 1 + 2.23e24T + 1.54e48T^{2} \)
5 \( 1 - 2.85e35T + 3.94e70T^{2} \)
7 \( 1 - 3.30e41T + 2.26e85T^{2} \)
11 \( 1 + 4.96e52T + 1.51e105T^{2} \)
13 \( 1 - 5.64e55T + 3.22e112T^{2} \)
17 \( 1 + 1.94e61T + 1.88e124T^{2} \)
19 \( 1 - 3.52e64T + 1.42e129T^{2} \)
23 \( 1 + 1.63e68T + 3.42e137T^{2} \)
29 \( 1 + 5.40e73T + 5.03e147T^{2} \)
31 \( 1 + 3.00e75T + 4.24e150T^{2} \)
37 \( 1 - 2.08e79T + 2.44e158T^{2} \)
41 \( 1 - 6.19e80T + 7.78e162T^{2} \)
43 \( 1 + 1.29e82T + 9.55e164T^{2} \)
47 \( 1 - 3.59e84T + 7.61e168T^{2} \)
53 \( 1 + 9.51e86T + 1.41e174T^{2} \)
59 \( 1 - 2.90e89T + 7.17e178T^{2} \)
61 \( 1 - 9.18e89T + 2.08e180T^{2} \)
67 \( 1 + 1.99e92T + 2.71e184T^{2} \)
71 \( 1 + 1.96e93T + 9.48e186T^{2} \)
73 \( 1 - 2.27e94T + 1.56e188T^{2} \)
79 \( 1 - 2.24e95T + 4.57e191T^{2} \)
83 \( 1 - 1.11e96T + 6.71e193T^{2} \)
89 \( 1 + 1.83e98T + 7.73e196T^{2} \)
97 \( 1 - 2.50e100T + 4.61e200T^{2} \)
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

−13.05726882317307182855147887329, −11.26648236080581207087359860845, −10.38300120215739225791751469515, −9.492443287025581014575019459167, −7.47377541199048210675539711983, −6.14685924467856461217161361031, −5.25247243719148982095164323036, −2.10150571005205108207446336184, −1.06379456386511640564012204306, 0, 1.06379456386511640564012204306, 2.10150571005205108207446336184, 5.25247243719148982095164323036, 6.14685924467856461217161361031, 7.47377541199048210675539711983, 9.492443287025581014575019459167, 10.38300120215739225791751469515, 11.26648236080581207087359860845, 13.05726882317307182855147887329

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