# Properties

 Degree 2 Conductor $2^{3} \cdot 3 \cdot 5 \cdot 67$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 3-s − 5-s + 4.52·7-s + 9-s + 3.01·11-s − 4.99·13-s − 15-s + 4.07·17-s + 1.78·19-s + 4.52·21-s + 4.12·23-s + 25-s + 27-s + 2.71·29-s + 4.66·31-s + 3.01·33-s − 4.52·35-s + 4.96·37-s − 4.99·39-s − 2.66·41-s − 9.11·43-s − 45-s − 2.72·47-s + 13.5·49-s + 4.07·51-s + 0.311·53-s − 3.01·55-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.447·5-s + 1.71·7-s + 0.333·9-s + 0.909·11-s − 1.38·13-s − 0.258·15-s + 0.987·17-s + 0.410·19-s + 0.988·21-s + 0.859·23-s + 0.200·25-s + 0.192·27-s + 0.503·29-s + 0.837·31-s + 0.525·33-s − 0.765·35-s + 0.816·37-s − 0.799·39-s − 0.415·41-s − 1.39·43-s − 0.149·45-s − 0.398·47-s + 1.92·49-s + 0.570·51-s + 0.0427·53-s − 0.406·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8040$$    =    $$2^{3} \cdot 3 \cdot 5 \cdot 67$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{8040} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 8040,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $3.372114619$ $L(\frac12)$ $\approx$ $3.372114619$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5,\;67\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1 - T$$
5 $$1 + T$$
67 $$1 - T$$
good7 $$1 - 4.52T + 7T^{2}$$
11 $$1 - 3.01T + 11T^{2}$$
13 $$1 + 4.99T + 13T^{2}$$
17 $$1 - 4.07T + 17T^{2}$$
19 $$1 - 1.78T + 19T^{2}$$
23 $$1 - 4.12T + 23T^{2}$$
29 $$1 - 2.71T + 29T^{2}$$
31 $$1 - 4.66T + 31T^{2}$$
37 $$1 - 4.96T + 37T^{2}$$
41 $$1 + 2.66T + 41T^{2}$$
43 $$1 + 9.11T + 43T^{2}$$
47 $$1 + 2.72T + 47T^{2}$$
53 $$1 - 0.311T + 53T^{2}$$
59 $$1 + 2.27T + 59T^{2}$$
61 $$1 + 0.999T + 61T^{2}$$
71 $$1 + 3.73T + 71T^{2}$$
73 $$1 - 13.6T + 73T^{2}$$
79 $$1 + 2.72T + 79T^{2}$$
83 $$1 + 8.69T + 83T^{2}$$
89 $$1 + 11.8T + 89T^{2}$$
97 $$1 - 8.55T + 97T^{2}$$
show less
\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

−7.965957204033953654083612296107, −7.27991287696727650508002886244, −6.72144733440786819724510852247, −5.54686844988109316746701328186, −4.80509230187638534131128542221, −4.48268099171924691011748181613, −3.46834827105846367734832791169, −2.68900605842701647950875201038, −1.71135429017082406503312239783, −0.960880405052537624384991842643, 0.960880405052537624384991842643, 1.71135429017082406503312239783, 2.68900605842701647950875201038, 3.46834827105846367734832791169, 4.48268099171924691011748181613, 4.80509230187638534131128542221, 5.54686844988109316746701328186, 6.72144733440786819724510852247, 7.27991287696727650508002886244, 7.965957204033953654083612296107