Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 89
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3.19e13·2-s + 3.23e21·3-s + 3.98e26·4-s + 1.00e30·5-s − 1.03e35·6-s + 1.60e37·7-s + 7.02e39·8-s + 7.55e42·9-s − 3.19e43·10-s − 3.40e46·11-s + 1.29e48·12-s − 5.21e49·13-s − 5.12e50·14-s + 3.24e51·15-s − 4.70e53·16-s − 3.62e53·17-s − 2.40e56·18-s − 5.12e56·19-s + 3.99e56·20-s + 5.19e58·21-s + 1.08e60·22-s − 4.07e60·23-s + 2.27e61·24-s − 1.60e62·25-s + 1.66e63·26-s + 1.50e64·27-s + 6.40e63·28-s + ⋯
L(s)  = 1  − 1.28·2-s + 1.89·3-s + 0.644·4-s + 0.0788·5-s − 2.43·6-s + 0.396·7-s + 0.456·8-s + 2.59·9-s − 0.101·10-s − 1.54·11-s + 1.22·12-s − 1.40·13-s − 0.508·14-s + 0.149·15-s − 1.22·16-s − 0.0636·17-s − 3.32·18-s − 0.638·19-s + 0.0508·20-s + 0.752·21-s + 1.98·22-s − 1.03·23-s + 0.864·24-s − 0.993·25-s + 1.79·26-s + 3.02·27-s + 0.255·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(90-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+89/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(89\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :89/2),\ -1)\)
\(L(45)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{91}{2})\)   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 \( 1 + 3.19e13T + 6.18e26T^{2} \)
3 \( 1 - 3.23e21T + 2.90e42T^{2} \)
5 \( 1 - 1.00e30T + 1.61e62T^{2} \)
7 \( 1 - 1.60e37T + 1.63e75T^{2} \)
11 \( 1 + 3.40e46T + 4.83e92T^{2} \)
13 \( 1 + 5.21e49T + 1.38e99T^{2} \)
17 \( 1 + 3.62e53T + 3.23e109T^{2} \)
19 \( 1 + 5.12e56T + 6.44e113T^{2} \)
23 \( 1 + 4.07e60T + 1.56e121T^{2} \)
29 \( 1 + 1.51e64T + 1.42e130T^{2} \)
31 \( 1 + 6.54e65T + 5.38e132T^{2} \)
37 \( 1 - 3.74e69T + 3.71e139T^{2} \)
41 \( 1 - 4.56e71T + 3.44e143T^{2} \)
43 \( 1 - 6.44e72T + 2.39e145T^{2} \)
47 \( 1 + 3.14e74T + 6.55e148T^{2} \)
53 \( 1 + 1.60e76T + 2.88e153T^{2} \)
59 \( 1 + 3.75e78T + 4.03e157T^{2} \)
61 \( 1 + 8.19e78T + 7.84e158T^{2} \)
67 \( 1 + 2.77e80T + 3.31e162T^{2} \)
71 \( 1 + 2.48e82T + 5.78e164T^{2} \)
73 \( 1 - 1.94e82T + 6.85e165T^{2} \)
79 \( 1 + 3.70e84T + 7.74e168T^{2} \)
83 \( 1 + 1.07e85T + 6.27e170T^{2} \)
89 \( 1 - 7.67e86T + 3.13e173T^{2} \)
97 \( 1 - 1.74e88T + 6.64e176T^{2} \)
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

−14.44136547315176701791382304144, −13.06702536168688140928599122971, −10.29724404125588318828635447451, −9.388889310186864466303721788017, −8.032579305763394114169991809704, −7.58586887612066082324723958034, −4.46793328136069584504190941734, −2.59148655381906762217474663683, −1.84015589183348535662397937312, 0, 1.84015589183348535662397937312, 2.59148655381906762217474663683, 4.46793328136069584504190941734, 7.58586887612066082324723958034, 8.032579305763394114169991809704, 9.388889310186864466303721788017, 10.29724404125588318828635447451, 13.06702536168688140928599122971, 14.44136547315176701791382304144

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