# Properties

 Degree 2 Conductor 5 Sign $-1$ Motivic weight 33 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.62e3·2-s − 1.24e8·3-s − 8.55e9·4-s − 1.52e11·5-s − 7.01e11·6-s + 7.01e13·7-s − 9.65e13·8-s + 9.96e15·9-s − 8.58e14·10-s + 9.39e16·11-s + 1.06e18·12-s + 9.95e17·13-s + 3.94e17·14-s + 1.90e19·15-s + 7.29e19·16-s − 2.59e20·17-s + 5.60e19·18-s − 1.31e21·19-s + 1.30e21·20-s − 8.74e21·21-s + 5.28e20·22-s + 5.44e22·23-s + 1.20e22·24-s + 2.32e22·25-s + 5.60e21·26-s − 5.49e23·27-s − 6.00e23·28-s + ⋯
 L(s)  = 1 + 0.0607·2-s − 1.67·3-s − 0.996·4-s − 0.447·5-s − 0.101·6-s + 0.797·7-s − 0.121·8-s + 1.79·9-s − 0.0271·10-s + 0.616·11-s + 1.66·12-s + 0.414·13-s + 0.0484·14-s + 0.747·15-s + 0.988·16-s − 1.29·17-s + 0.108·18-s − 1.04·19-s + 0.445·20-s − 1.33·21-s + 0.0374·22-s + 1.85·23-s + 0.202·24-s + 0.200·25-s + 0.0251·26-s − 1.32·27-s − 0.794·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$5$$ $$\varepsilon$$ = $-1$ motivic weight = $$33$$ character : $\chi_{5} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 5,\ (\ :33/2),\ -1)$ $L(17)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{35}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $$F_p$$ is a polynomial of degree 2. If $p = 5$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 $$1 + 1.52e11T$$
good2 $$1 - 5.62e3T + 8.58e9T^{2}$$
3 $$1 + 1.24e8T + 5.55e15T^{2}$$
7 $$1 - 7.01e13T + 7.73e27T^{2}$$
11 $$1 - 9.39e16T + 2.32e34T^{2}$$
13 $$1 - 9.95e17T + 5.75e36T^{2}$$
17 $$1 + 2.59e20T + 4.02e40T^{2}$$
19 $$1 + 1.31e21T + 1.58e42T^{2}$$
23 $$1 - 5.44e22T + 8.65e44T^{2}$$
29 $$1 + 9.77e23T + 1.81e48T^{2}$$
31 $$1 - 6.82e24T + 1.64e49T^{2}$$
37 $$1 + 7.50e25T + 5.63e51T^{2}$$
41 $$1 + 2.94e25T + 1.66e53T^{2}$$
43 $$1 + 1.00e27T + 8.02e53T^{2}$$
47 $$1 - 1.91e27T + 1.51e55T^{2}$$
53 $$1 - 4.33e28T + 7.96e56T^{2}$$
59 $$1 + 1.21e29T + 2.74e58T^{2}$$
61 $$1 - 4.03e29T + 8.23e58T^{2}$$
67 $$1 + 1.33e30T + 1.82e60T^{2}$$
71 $$1 + 4.10e30T + 1.23e61T^{2}$$
73 $$1 - 8.64e30T + 3.08e61T^{2}$$
79 $$1 + 1.76e31T + 4.18e62T^{2}$$
83 $$1 + 6.43e30T + 2.13e63T^{2}$$
89 $$1 - 1.32e32T + 2.13e64T^{2}$$
97 $$1 - 2.90e32T + 3.65e65T^{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

−15.13750264624874768371020073832, −13.18044754966314464041737505683, −11.75705240373532426038355083913, −10.69383533862968044208426783688, −8.701726712481703178513907096448, −6.65713411629284494211826183269, −5.09588363391465376274668579748, −4.22078326040349388487576387208, −1.15229297651253518036906482209, 0, 1.15229297651253518036906482209, 4.22078326040349388487576387208, 5.09588363391465376274668579748, 6.65713411629284494211826183269, 8.701726712481703178513907096448, 10.69383533862968044208426783688, 11.75705240373532426038355083913, 13.18044754966314464041737505683, 15.13750264624874768371020073832