# Properties

 Label 2-338-1.1-c5-0-29 Degree $2$ Conductor $338$ Sign $1$ Analytic cond. $54.2097$ Root an. cond. $7.36272$ Motivic weight $5$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 4·2-s + 16·4-s + 14·5-s + 170·7-s + 64·8-s − 243·9-s + 56·10-s + 250·11-s + 680·14-s + 256·16-s + 1.06e3·17-s − 972·18-s + 78·19-s + 224·20-s + 1.00e3·22-s + 1.57e3·23-s − 2.92e3·25-s + 2.72e3·28-s + 2.57e3·29-s + 8.65e3·31-s + 1.02e3·32-s + 4.24e3·34-s + 2.38e3·35-s − 3.88e3·36-s − 1.09e4·37-s + 312·38-s + 896·40-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/2·4-s + 0.250·5-s + 1.31·7-s + 0.353·8-s − 9-s + 0.177·10-s + 0.622·11-s + 0.927·14-s + 1/4·16-s + 0.891·17-s − 0.707·18-s + 0.0495·19-s + 0.125·20-s + 0.440·22-s + 0.621·23-s − 0.937·25-s + 0.655·28-s + 0.569·29-s + 1.61·31-s + 0.176·32-s + 0.630·34-s + 0.328·35-s − 1/2·36-s − 1.31·37-s + 0.0350·38-s + 0.0885·40-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$338$$    =    $$2 \cdot 13^{2}$$ Sign: $1$ Analytic conductor: $$54.2097$$ Root analytic conductor: $$7.36272$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 338,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$4.210355428$$ $$L(\frac12)$$ $$\approx$$ $$4.210355428$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - p^{2} T$$
13 $$1$$
good3 $$1 + p^{5} T^{2}$$
5 $$1 - 14 T + p^{5} T^{2}$$
7 $$1 - 170 T + p^{5} T^{2}$$
11 $$1 - 250 T + p^{5} T^{2}$$
17 $$1 - 1062 T + p^{5} T^{2}$$
19 $$1 - 78 T + p^{5} T^{2}$$
23 $$1 - 1576 T + p^{5} T^{2}$$
29 $$1 - 2578 T + p^{5} T^{2}$$
31 $$1 - 8654 T + p^{5} T^{2}$$
37 $$1 + 10986 T + p^{5} T^{2}$$
41 $$1 + 1050 T + p^{5} T^{2}$$
43 $$1 + 5900 T + p^{5} T^{2}$$
47 $$1 - 5962 T + p^{5} T^{2}$$
53 $$1 - 29046 T + p^{5} T^{2}$$
59 $$1 - 13922 T + p^{5} T^{2}$$
61 $$1 + 32882 T + p^{5} T^{2}$$
67 $$1 - 69566 T + p^{5} T^{2}$$
71 $$1 - 50542 T + p^{5} T^{2}$$
73 $$1 - 46750 T + p^{5} T^{2}$$
79 $$1 + 19348 T + p^{5} T^{2}$$
83 $$1 - 87438 T + p^{5} T^{2}$$
89 $$1 + 94170 T + p^{5} T^{2}$$
97 $$1 + 182786 T + p^{5} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$