# Properties

 Degree 2 Conductor 5 Sign $1$ Motivic weight 33 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.26e5·2-s − 8.63e7·3-s + 7.38e9·4-s + 1.52e11·5-s + 1.09e13·6-s + 7.78e13·7-s + 1.52e14·8-s + 1.89e15·9-s − 1.92e16·10-s + 1.61e17·11-s − 6.37e17·12-s − 2.28e18·13-s − 9.84e18·14-s − 1.31e19·15-s − 8.26e19·16-s + 2.46e20·17-s − 2.39e20·18-s − 1.53e21·19-s + 1.12e21·20-s − 6.72e21·21-s − 2.04e22·22-s − 2.16e22·23-s − 1.31e22·24-s + 2.32e22·25-s + 2.89e23·26-s + 3.16e23·27-s + 5.75e23·28-s + ⋯
 L(s)  = 1 − 1.36·2-s − 1.15·3-s + 0.859·4-s + 0.447·5-s + 1.57·6-s + 0.885·7-s + 0.190·8-s + 0.340·9-s − 0.609·10-s + 1.06·11-s − 0.995·12-s − 0.953·13-s − 1.20·14-s − 0.517·15-s − 1.12·16-s + 1.22·17-s − 0.464·18-s − 1.22·19-s + 0.384·20-s − 1.02·21-s − 1.44·22-s − 0.735·23-s − 0.221·24-s + 0.200·25-s + 1.30·26-s + 0.763·27-s + 0.761·28-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 = 0 Selberg data = $(2,\ 5,\ (\ :33/2),\ 1)$ $L(17)$ $\approx$ $0.6489982189$ $L(\frac12)$ $\approx$ $0.6489982189$ $L(\frac{35}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 5$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 5$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 $$1 - 1.52e11T$$
good2 $$1 + 1.26e5T + 8.58e9T^{2}$$
3 $$1 + 8.63e7T + 5.55e15T^{2}$$
7 $$1 - 7.78e13T + 7.73e27T^{2}$$
11 $$1 - 1.61e17T + 2.32e34T^{2}$$
13 $$1 + 2.28e18T + 5.75e36T^{2}$$
17 $$1 - 2.46e20T + 4.02e40T^{2}$$
19 $$1 + 1.53e21T + 1.58e42T^{2}$$
23 $$1 + 2.16e22T + 8.65e44T^{2}$$
29 $$1 + 2.07e23T + 1.81e48T^{2}$$
31 $$1 - 1.75e24T + 1.64e49T^{2}$$
37 $$1 - 8.12e25T + 5.63e51T^{2}$$
41 $$1 + 1.92e26T + 1.66e53T^{2}$$
43 $$1 - 8.01e26T + 8.02e53T^{2}$$
47 $$1 + 7.53e27T + 1.51e55T^{2}$$
53 $$1 - 1.75e28T + 7.96e56T^{2}$$
59 $$1 + 2.97e29T + 2.74e58T^{2}$$
61 $$1 + 3.70e29T + 8.23e58T^{2}$$
67 $$1 - 8.43e29T + 1.82e60T^{2}$$
71 $$1 - 5.92e30T + 1.23e61T^{2}$$
73 $$1 - 7.49e30T + 3.08e61T^{2}$$
79 $$1 - 6.07e30T + 4.18e62T^{2}$$
83 $$1 - 3.47e31T + 2.13e63T^{2}$$
89 $$1 - 1.80e32T + 2.13e64T^{2}$$
97 $$1 - 1.09e33T + 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

−16.76199098162569760365475918922, −14.44994961692951512737681442002, −12.05305686940475542548586740387, −10.85245258198327515084305817331, −9.607200677611740859295500562729, −8.002717772307227546656617874337, −6.36637790558535352432706740019, −4.75978964464573230976129542578, −1.81880437593070529066471579048, −0.66056961295763754977032736231, 0.66056961295763754977032736231, 1.81880437593070529066471579048, 4.75978964464573230976129542578, 6.36637790558535352432706740019, 8.002717772307227546656617874337, 9.607200677611740859295500562729, 10.85245258198327515084305817331, 12.05305686940475542548586740387, 14.44994961692951512737681442002, 16.76199098162569760365475918922