# Properties

 Degree 2 Conductor 2 Sign $-1$ Motivic weight 81 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.09e12·2-s + 2.69e19·3-s + 1.20e24·4-s + 1.65e28·5-s − 2.96e31·6-s + 2.15e33·7-s − 1.32e36·8-s + 2.82e38·9-s − 1.82e40·10-s − 3.85e40·11-s + 3.25e43·12-s − 8.46e43·13-s − 2.36e45·14-s + 4.46e47·15-s + 1.46e48·16-s − 7.68e49·17-s − 3.10e50·18-s − 7.37e51·19-s + 2.00e52·20-s + 5.80e52·21-s + 4.23e52·22-s − 1.05e55·23-s − 3.58e55·24-s − 1.39e56·25-s + 9.31e55·26-s − 4.34e57·27-s + 2.60e57·28-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1.27·3-s + 0.5·4-s + 0.814·5-s − 0.904·6-s + 0.127·7-s − 0.353·8-s + 0.636·9-s − 0.575·10-s − 0.0256·11-s + 0.639·12-s − 0.0650·13-s − 0.0904·14-s + 1.04·15-s + 0.250·16-s − 1.12·17-s − 0.450·18-s − 1.19·19-s + 0.407·20-s + 0.163·21-s + 0.0181·22-s − 0.745·23-s − 0.452·24-s − 0.337·25-s + 0.0459·26-s − 0.465·27-s + 0.0639·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(82-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+81/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2$$ $$\varepsilon$$ = $-1$ motivic weight = $$81$$ character : $\chi_{2} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 2,\ (\ :81/2),\ -1)$ $L(41)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{83}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + 1.09e12T$$
good3 $$1 - 2.69e19T + 4.43e38T^{2}$$
5 $$1 - 1.65e28T + 4.13e56T^{2}$$
7 $$1 - 2.15e33T + 2.83e68T^{2}$$
11 $$1 + 3.85e40T + 2.25e84T^{2}$$
13 $$1 + 8.46e43T + 1.69e90T^{2}$$
17 $$1 + 7.68e49T + 4.63e99T^{2}$$
19 $$1 + 7.37e51T + 3.79e103T^{2}$$
23 $$1 + 1.05e55T + 1.99e110T^{2}$$
29 $$1 + 1.79e59T + 2.84e118T^{2}$$
31 $$1 + 4.86e60T + 6.31e120T^{2}$$
37 $$1 - 2.30e63T + 1.05e127T^{2}$$
41 $$1 + 1.35e65T + 4.32e130T^{2}$$
43 $$1 - 2.10e66T + 2.04e132T^{2}$$
47 $$1 - 1.03e68T + 2.75e135T^{2}$$
53 $$1 - 2.99e69T + 4.63e139T^{2}$$
59 $$1 + 5.68e71T + 2.74e143T^{2}$$
61 $$1 - 5.44e71T + 4.08e144T^{2}$$
67 $$1 + 5.75e73T + 8.16e147T^{2}$$
71 $$1 - 4.64e74T + 8.95e149T^{2}$$
73 $$1 - 4.70e75T + 8.49e150T^{2}$$
79 $$1 + 4.20e76T + 5.10e153T^{2}$$
83 $$1 + 7.46e77T + 2.78e155T^{2}$$
89 $$1 - 5.59e78T + 7.95e157T^{2}$$
97 $$1 + 1.28e80T + 8.48e160T^{2}$$
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\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

−12.96446727965251858602492250770, −10.91540996963695945676992130567, −9.494518375178867106013539250488, −8.723154818811695291405901533350, −7.47186979821420211745410067144, −5.94217770458994949981878820727, −3.93042105477405562264675147715, −2.36424398822134285468909051906, −1.85240261465840691084710946396, 0, 1.85240261465840691084710946396, 2.36424398822134285468909051906, 3.93042105477405562264675147715, 5.94217770458994949981878820727, 7.47186979821420211745410067144, 8.723154818811695291405901533350, 9.494518375178867106013539250488, 10.91540996963695945676992130567, 12.96446727965251858602492250770