Properties

 Degree $2$ Conductor $25$ Sign $-1$ Motivic weight $9$ Primitive yes Self-dual yes Analytic rank $1$

Related objects

Dirichlet series

 L(s)  = 1 + 20.2·2-s − 30.5·3-s − 103.·4-s − 616.·6-s + 4.01e3·7-s − 1.24e4·8-s − 1.87e4·9-s − 4.21e4·11-s + 3.16e3·12-s − 1.23e5·13-s + 8.10e4·14-s − 1.98e5·16-s − 3.19e5·17-s − 3.78e5·18-s + 1.08e6·19-s − 1.22e5·21-s − 8.50e5·22-s − 1.50e6·23-s + 3.79e5·24-s − 2.49e6·26-s + 1.17e6·27-s − 4.16e5·28-s − 2.62e6·29-s + 3.27e6·31-s + 2.36e6·32-s + 1.28e6·33-s − 6.46e6·34-s + ⋯
 L(s)  = 1 + 0.892·2-s − 0.217·3-s − 0.202·4-s − 0.194·6-s + 0.631·7-s − 1.07·8-s − 0.952·9-s − 0.867·11-s + 0.0441·12-s − 1.20·13-s + 0.563·14-s − 0.755·16-s − 0.929·17-s − 0.850·18-s + 1.91·19-s − 0.137·21-s − 0.774·22-s − 1.12·23-s + 0.233·24-s − 1.07·26-s + 0.424·27-s − 0.128·28-s − 0.688·29-s + 0.635·31-s + 0.399·32-s + 0.188·33-s − 0.829·34-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$25$$    =    $$5^{2}$$ Sign: $-1$ Motivic weight: $$9$$ Character: $\chi_{25} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 25,\ (\ :9/2),\ -1)$$

Particular Values

 $$L(5)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
good2 $$1 - 20.2T + 512T^{2}$$
3 $$1 + 30.5T + 1.96e4T^{2}$$
7 $$1 - 4.01e3T + 4.03e7T^{2}$$
11 $$1 + 4.21e4T + 2.35e9T^{2}$$
13 $$1 + 1.23e5T + 1.06e10T^{2}$$
17 $$1 + 3.19e5T + 1.18e11T^{2}$$
19 $$1 - 1.08e6T + 3.22e11T^{2}$$
23 $$1 + 1.50e6T + 1.80e12T^{2}$$
29 $$1 + 2.62e6T + 1.45e13T^{2}$$
31 $$1 - 3.27e6T + 2.64e13T^{2}$$
37 $$1 - 2.51e6T + 1.29e14T^{2}$$
41 $$1 - 2.95e7T + 3.27e14T^{2}$$
43 $$1 - 1.42e7T + 5.02e14T^{2}$$
47 $$1 + 1.35e6T + 1.11e15T^{2}$$
53 $$1 + 9.73e7T + 3.29e15T^{2}$$
59 $$1 + 7.48e6T + 8.66e15T^{2}$$
61 $$1 + 9.11e7T + 1.16e16T^{2}$$
67 $$1 + 2.94e8T + 2.72e16T^{2}$$
71 $$1 - 1.56e8T + 4.58e16T^{2}$$
73 $$1 - 2.82e8T + 5.88e16T^{2}$$
79 $$1 + 5.55e8T + 1.19e17T^{2}$$
83 $$1 + 6.48e6T + 1.86e17T^{2}$$
89 $$1 + 5.99e8T + 3.50e17T^{2}$$
97 $$1 + 9.25e8T + 7.60e17T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$