# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 7 \cdot 11$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.58·2-s + 3-s + 4.70·4-s − 5-s − 2.58·6-s − 7-s − 6.99·8-s + 9-s + 2.58·10-s − 11-s + 4.70·12-s − 2.40·13-s + 2.58·14-s − 15-s + 8.70·16-s − 3.70·17-s − 2.58·18-s + 4.15·19-s − 4.70·20-s − 21-s + 2.58·22-s − 0.0692·23-s − 6.99·24-s + 25-s + 6.22·26-s + 27-s − 4.70·28-s + ⋯
 L(s)  = 1 − 1.83·2-s + 0.577·3-s + 2.35·4-s − 0.447·5-s − 1.05·6-s − 0.377·7-s − 2.47·8-s + 0.333·9-s + 0.818·10-s − 0.301·11-s + 1.35·12-s − 0.666·13-s + 0.691·14-s − 0.258·15-s + 2.17·16-s − 0.897·17-s − 0.610·18-s + 0.953·19-s − 1.05·20-s − 0.218·21-s + 0.551·22-s − 0.0144·23-s − 1.42·24-s + 0.200·25-s + 1.22·26-s + 0.192·27-s − 0.888·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1155$$    =    $$3 \cdot 5 \cdot 7 \cdot 11$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{1155} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 1155,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$0.6591865022$$ $$L(\frac12)$$ $$\approx$$ $$0.6591865022$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;7,\;11\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 - T$$
5 $$1 + T$$
7 $$1 + T$$
11 $$1 + T$$
good2 $$1 + 2.58T + 2T^{2}$$
13 $$1 + 2.40T + 13T^{2}$$
17 $$1 + 3.70T + 17T^{2}$$
19 $$1 - 4.15T + 19T^{2}$$
23 $$1 + 0.0692T + 23T^{2}$$
29 $$1 - 0.703T + 29T^{2}$$
31 $$1 - 5.22T + 31T^{2}$$
37 $$1 - 7.95T + 37T^{2}$$
41 $$1 - 2.31T + 41T^{2}$$
43 $$1 - 4.24T + 43T^{2}$$
47 $$1 - 13.3T + 47T^{2}$$
53 $$1 + 8.51T + 53T^{2}$$
59 $$1 + 4.47T + 59T^{2}$$
61 $$1 - 5.02T + 61T^{2}$$
67 $$1 - 4.72T + 67T^{2}$$
71 $$1 - 4.40T + 71T^{2}$$
73 $$1 + 14.2T + 73T^{2}$$
79 $$1 + 4.40T + 79T^{2}$$
83 $$1 - 5.56T + 83T^{2}$$
89 $$1 - 15.1T + 89T^{2}$$
97 $$1 + 8.24T + 97T^{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

−9.569215425675887458666642814400, −9.078817395623990327795389195183, −8.213607569143270422488818730559, −7.57436886988685560740685531690, −6.99124926173308188664323094612, −5.99669726581949430975170155912, −4.47224355619173964652126025914, −3.03622844816130731329806270061, −2.26836298564414965110307697230, −0.76246366574970015371563127252, 0.76246366574970015371563127252, 2.26836298564414965110307697230, 3.03622844816130731329806270061, 4.47224355619173964652126025914, 5.99669726581949430975170155912, 6.99124926173308188664323094612, 7.57436886988685560740685531690, 8.213607569143270422488818730559, 9.078817395623990327795389195183, 9.569215425675887458666642814400