# Properties

 Degree $2$ Conductor $53$ Sign $-1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 3·3-s − 4-s + 3·6-s − 4·7-s + 3·8-s + 6·9-s + 3·12-s − 3·13-s + 4·14-s − 16-s − 3·17-s − 6·18-s − 5·19-s + 12·21-s + 7·23-s − 9·24-s − 5·25-s + 3·26-s − 9·27-s + 4·28-s − 7·29-s + 4·31-s − 5·32-s + 3·34-s − 6·36-s + 5·37-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.73·3-s − 1/2·4-s + 1.22·6-s − 1.51·7-s + 1.06·8-s + 2·9-s + 0.866·12-s − 0.832·13-s + 1.06·14-s − 1/4·16-s − 0.727·17-s − 1.41·18-s − 1.14·19-s + 2.61·21-s + 1.45·23-s − 1.83·24-s − 25-s + 0.588·26-s − 1.73·27-s + 0.755·28-s − 1.29·29-s + 0.718·31-s − 0.883·32-s + 0.514·34-s − 36-s + 0.821·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$53$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{53} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 53,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad53 $$1 + T$$
good2 $$1 + T + p T^{2}$$
3 $$1 + p T + p T^{2}$$
5 $$1 + p T^{2}$$
7 $$1 + 4 T + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 + 3 T + p T^{2}$$
17 $$1 + 3 T + p T^{2}$$
19 $$1 + 5 T + p T^{2}$$
23 $$1 - 7 T + p T^{2}$$
29 $$1 + 7 T + p T^{2}$$
31 $$1 - 4 T + p T^{2}$$
37 $$1 - 5 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 + 2 T + p T^{2}$$
47 $$1 + 2 T + p T^{2}$$
59 $$1 + 2 T + p T^{2}$$
61 $$1 + 8 T + p T^{2}$$
67 $$1 + 12 T + p T^{2}$$
71 $$1 - T + p T^{2}$$
73 $$1 + 4 T + p T^{2}$$
79 $$1 + T + p T^{2}$$
83 $$1 + T + p T^{2}$$
89 $$1 + 14 T + p T^{2}$$
97 $$1 - T + p T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−19.55130022540896, −18.96307855314917, −17.87397635420920, −16.99882325261332, −16.56634537954273, −15.32142426611891, −13.21929603222722, −12.62142615234457, −11.17082899287976, −10.14893340042306, −9.264238826193943, −7.207369479992710, −6.043478899410247, −4.508628350682962, 0, 4.508628350682962, 6.043478899410247, 7.207369479992710, 9.264238826193943, 10.14893340042306, 11.17082899287976, 12.62142615234457, 13.21929603222722, 15.32142426611891, 16.56634537954273, 16.99882325261332, 17.87397635420920, 18.96307855314917, 19.55130022540896