Properties

 Label 2-95e2-1.1-c1-0-418 Degree $2$ Conductor $9025$ Sign $-1$ Analytic cond. $72.0649$ Root an. cond. $8.48910$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

Related objects

Dirichlet series

 L(s)  = 1 + 1.17·2-s + 1.17·3-s − 0.618·4-s + 1.38·6-s − 0.236·7-s − 3.07·8-s − 1.61·9-s + 0.854·11-s − 0.726·12-s + 0.726·13-s − 0.277·14-s − 2.38·16-s − 0.763·17-s − 1.90·18-s − 0.277·21-s + 1.00·22-s + 7.09·23-s − 3.61·24-s + 0.854·26-s − 5.42·27-s + 0.145·28-s + 8.78·29-s + 1.17·31-s + 3.35·32-s + 1.00·33-s − 0.898·34-s + 1.00·36-s + ⋯
 L(s)  = 1 + 0.831·2-s + 0.678·3-s − 0.309·4-s + 0.564·6-s − 0.0892·7-s − 1.08·8-s − 0.539·9-s + 0.257·11-s − 0.209·12-s + 0.201·13-s − 0.0741·14-s − 0.595·16-s − 0.185·17-s − 0.448·18-s − 0.0605·21-s + 0.214·22-s + 1.47·23-s − 0.738·24-s + 0.167·26-s − 1.04·27-s + 0.0275·28-s + 1.63·29-s + 0.211·31-s + 0.593·32-s + 0.174·33-s − 0.154·34-s + 0.166·36-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$9025$$    =    $$5^{2} \cdot 19^{2}$$ Sign: $-1$ Analytic conductor: $$72.0649$$ Root analytic conductor: $$8.48910$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 9025,\ (\ :1/2),\ -1)$$

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$$F_p(T)$
bad5 $$1$$
19 $$1$$
good2 $$1 - 1.17T + 2T^{2}$$
3 $$1 - 1.17T + 3T^{2}$$
7 $$1 + 0.236T + 7T^{2}$$
11 $$1 - 0.854T + 11T^{2}$$
13 $$1 - 0.726T + 13T^{2}$$
17 $$1 + 0.763T + 17T^{2}$$
23 $$1 - 7.09T + 23T^{2}$$
29 $$1 - 8.78T + 29T^{2}$$
31 $$1 - 1.17T + 31T^{2}$$
37 $$1 + 8.78T + 37T^{2}$$
41 $$1 + 1.62T + 41T^{2}$$
43 $$1 + 2.61T + 43T^{2}$$
47 $$1 + 7.47T + 47T^{2}$$
53 $$1 - 1.00T + 53T^{2}$$
59 $$1 - 11.3T + 59T^{2}$$
61 $$1 + 13.9T + 61T^{2}$$
67 $$1 + 11.5T + 67T^{2}$$
71 $$1 + 13.0T + 71T^{2}$$
73 $$1 - T + 73T^{2}$$
79 $$1 + 8.50T + 79T^{2}$$
83 $$1 - 13.2T + 83T^{2}$$
89 $$1 + 8.33T + 89T^{2}$$
97 $$1 - 4.25T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$