# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.60·2-s + 4.78·4-s + 4.31·5-s − 7-s + 7.23·8-s + 11.2·10-s + 2.46·13-s − 2.60·14-s + 9.29·16-s − 1.18·17-s + 3.42·19-s + 20.6·20-s − 8.32·23-s + 13.6·25-s + 6.41·26-s − 4.78·28-s + 1.45·29-s + 2.98·31-s + 9.71·32-s − 3.08·34-s − 4.31·35-s + 6.34·37-s + 8.92·38-s + 31.2·40-s − 6.16·41-s − 3.15·43-s − 21.6·46-s + ⋯
 L(s)  = 1 + 1.84·2-s + 2.39·4-s + 1.93·5-s − 0.377·7-s + 2.55·8-s + 3.55·10-s + 0.683·13-s − 0.695·14-s + 2.32·16-s − 0.287·17-s + 0.786·19-s + 4.61·20-s − 1.73·23-s + 2.73·25-s + 1.25·26-s − 0.903·28-s + 0.269·29-s + 0.536·31-s + 1.71·32-s − 0.529·34-s − 0.729·35-s + 1.04·37-s + 1.44·38-s + 4.94·40-s − 0.962·41-s − 0.480·43-s − 3.19·46-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$7623$$    =    $$3^{2} \cdot 7 \cdot 11^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{7623} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 7623,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $10.23236668$ $L(\frac12)$ $\approx$ $10.23236668$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;7,\;11\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + T$$
11 $$1$$
good2 $$1 - 2.60T + 2T^{2}$$
5 $$1 - 4.31T + 5T^{2}$$
13 $$1 - 2.46T + 13T^{2}$$
17 $$1 + 1.18T + 17T^{2}$$
19 $$1 - 3.42T + 19T^{2}$$
23 $$1 + 8.32T + 23T^{2}$$
29 $$1 - 1.45T + 29T^{2}$$
31 $$1 - 2.98T + 31T^{2}$$
37 $$1 - 6.34T + 37T^{2}$$
41 $$1 + 6.16T + 41T^{2}$$
43 $$1 + 3.15T + 43T^{2}$$
47 $$1 + 8.73T + 47T^{2}$$
53 $$1 - 6.68T + 53T^{2}$$
59 $$1 + 11.7T + 59T^{2}$$
61 $$1 + 7.76T + 61T^{2}$$
67 $$1 - 3.11T + 67T^{2}$$
71 $$1 + 5.48T + 71T^{2}$$
73 $$1 + 8.66T + 73T^{2}$$
79 $$1 - 8.83T + 79T^{2}$$
83 $$1 + 9.66T + 83T^{2}$$
89 $$1 + 0.985T + 89T^{2}$$
97 $$1 - 6.97T + 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

−7.51900503828649394355083112699, −6.64844196405367210642320020009, −6.07314018483608838530808611512, −5.96553406263677730007712425683, −5.07486280643750122434109152699, −4.50776201765889904366122806142, −3.49370637109205258610071642854, −2.85181636006035773028079371244, −2.07435248470901871098373589998, −1.40661473689758627434928155529, 1.40661473689758627434928155529, 2.07435248470901871098373589998, 2.85181636006035773028079371244, 3.49370637109205258610071642854, 4.50776201765889904366122806142, 5.07486280643750122434109152699, 5.96553406263677730007712425683, 6.07314018483608838530808611512, 6.64844196405367210642320020009, 7.51900503828649394355083112699