# Properties

 Degree 2 Conductor $7 \cdot 11 \cdot 43$ Sign $1$ Motivic weight 0 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 2·2-s + 3·4-s + 7-s + 4·8-s − 9-s − 11-s + 2·14-s + 5·16-s − 2·18-s − 2·22-s − 2·23-s − 25-s + 3·28-s − 2·29-s + 6·32-s − 3·36-s − 43-s − 3·44-s − 4·46-s + 49-s − 2·50-s + 2·53-s + 4·56-s − 4·58-s − 63-s + 7·64-s + 2·67-s + ⋯
 L(s)  = 1 + 2·2-s + 3·4-s + 7-s + 4·8-s − 9-s − 11-s + 2·14-s + 5·16-s − 2·18-s − 2·22-s − 2·23-s − 25-s + 3·28-s − 2·29-s + 6·32-s − 3·36-s − 43-s − 3·44-s − 4·46-s + 49-s − 2·50-s + 2·53-s + 4·56-s − 4·58-s − 63-s + 7·64-s + 2·67-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3311$$    =    $$7 \cdot 11 \cdot 43$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : $\chi_{3311} (3310, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 3311,\ (\ :0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $4.436512507$ $L(\frac12)$ $\approx$ $4.436512507$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{7,\;11,\;43\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{7,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 $$1 - T$$
11 $$1 + T$$
43 $$1 + T$$
good2 $$( 1 - T )^{2}$$
3 $$1 + T^{2}$$
5 $$1 + T^{2}$$
13 $$1 + T^{2}$$
17 $$1 + T^{2}$$
19 $$( 1 - T )( 1 + T )$$
23 $$( 1 + T )^{2}$$
29 $$( 1 + T )^{2}$$
31 $$( 1 - T )( 1 + T )$$
37 $$( 1 - T )( 1 + T )$$
41 $$1 + T^{2}$$
47 $$( 1 - T )( 1 + T )$$
53 $$( 1 - T )^{2}$$
59 $$( 1 - T )( 1 + T )$$
61 $$( 1 - T )( 1 + T )$$
67 $$( 1 - T )^{2}$$
71 $$( 1 - T )( 1 + T )$$
73 $$( 1 - T )( 1 + T )$$
79 $$( 1 - T )( 1 + T )$$
83 $$1 + T^{2}$$
89 $$1 + T^{2}$$
97 $$( 1 - T )( 1 + T )$$
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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

−8.327529048966035595447667808121, −7.86576770894213139238402564395, −7.21276800265659617620724937401, −6.10546057474397191428028034742, −5.55683797397808236356124878961, −5.16058185068508082676236724432, −4.12271596725253729989644227068, −3.56122613556173330407350824039, −2.37195125549253654885749496071, −1.94810746962855116347226891179, 1.94810746962855116347226891179, 2.37195125549253654885749496071, 3.56122613556173330407350824039, 4.12271596725253729989644227068, 5.16058185068508082676236724432, 5.55683797397808236356124878961, 6.10546057474397191428028034742, 7.21276800265659617620724937401, 7.86576770894213139238402564395, 8.327529048966035595447667808121