# Properties

 Degree 2 Conductor $2^{2} \cdot 29$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·3-s − 2·5-s + 4·7-s + 9-s − 6·11-s + 2·13-s − 4·15-s + 2·17-s − 6·19-s + 8·21-s + 4·23-s − 25-s − 4·27-s − 29-s − 6·31-s − 12·33-s − 8·35-s + 2·37-s + 4·39-s + 2·41-s + 10·43-s − 2·45-s − 2·47-s + 9·49-s + 4·51-s + 10·53-s + 12·55-s + ⋯
 L(s)  = 1 + 1.15·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 1.03·15-s + 0.485·17-s − 1.37·19-s + 1.74·21-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 0.185·29-s − 1.07·31-s − 2.08·33-s − 1.35·35-s + 0.328·37-s + 0.640·39-s + 0.312·41-s + 1.52·43-s − 0.298·45-s − 0.291·47-s + 9/7·49-s + 0.560·51-s + 1.37·53-s + 1.61·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$116$$    =    $$2^{2} \cdot 29$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{116} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 116,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.323119098$ $L(\frac12)$ $\approx$ $1.323119098$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;29\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
29 $$1 + T$$
good3 $$1 - 2 T + p T^{2}$$
5 $$1 + 2 T + p T^{2}$$
7 $$1 - 4 T + p T^{2}$$
11 $$1 + 6 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 + 6 T + p T^{2}$$
23 $$1 - 4 T + p T^{2}$$
31 $$1 + 6 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 - 2 T + p T^{2}$$
43 $$1 - 10 T + p T^{2}$$
47 $$1 + 2 T + p T^{2}$$
53 $$1 - 10 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 + 12 T + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 - 10 T + p T^{2}$$
79 $$1 + 6 T + p T^{2}$$
83 $$1 - 16 T + p T^{2}$$
89 $$1 - 2 T + p T^{2}$$
97 $$1 - 10 T + p T^{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

−19.20853250133328, −18.51427961106496, −17.54613534341227, −16.20205000493621, −15.12474620287817, −14.79172056037881, −13.64913618416886, −12.70905976606632, −11.29976427066418, −10.62296992663263, −8.881475557679452, −8.042165076629360, −7.613462457402513, −5.332743547825179, −3.964897420015663, −2.388482371431609, 2.388482371431609, 3.964897420015663, 5.332743547825179, 7.613462457402513, 8.042165076629360, 8.881475557679452, 10.62296992663263, 11.29976427066418, 12.70905976606632, 13.64913618416886, 14.79172056037881, 15.12474620287817, 16.20205000493621, 17.54613534341227, 18.51427961106496, 19.20853250133328