# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $-0.997 + 0.0684i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.20 − 1.27i)3-s + (3.56 − 2.05i)5-s + (−6.98 − 0.407i)7-s + (−1.27 − 2.20i)9-s + (−1.63 + 2.83i)11-s + 5.88i·13-s − 10.4·15-s + (−12.0 − 6.93i)17-s + (−13.7 + 7.91i)19-s + (14.8 + 9.77i)21-s + (−18.2 − 31.6i)23-s + (−4.01 + 6.95i)25-s + 29.3i·27-s − 28.4·29-s + (−36.2 − 20.9i)31-s + ⋯
 L(s)  = 1 + (−0.733 − 0.423i)3-s + (0.713 − 0.411i)5-s + (−0.998 − 0.0581i)7-s + (−0.141 − 0.244i)9-s + (−0.148 + 0.257i)11-s + 0.452i·13-s − 0.697·15-s + (−0.707 − 0.408i)17-s + (−0.721 + 0.416i)19-s + (0.707 + 0.465i)21-s + (−0.793 − 1.37i)23-s + (−0.160 + 0.278i)25-s + 1.08i·27-s − 0.981·29-s + (−1.16 − 0.674i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0684i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $-0.997 + 0.0684i$ motivic weight = $$2$$ character : $\chi_{224} (129, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :1),\ -0.997 + 0.0684i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.0127623 - 0.372295i$$ $$L(\frac12)$$ $$\approx$$ $$0.0127623 - 0.372295i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
7 $$1 + (6.98 + 0.407i)T$$
good3 $$1 + (2.20 + 1.27i)T + (4.5 + 7.79i)T^{2}$$
5 $$1 + (-3.56 + 2.05i)T + (12.5 - 21.6i)T^{2}$$
11 $$1 + (1.63 - 2.83i)T + (-60.5 - 104. i)T^{2}$$
13 $$1 - 5.88iT - 169T^{2}$$
17 $$1 + (12.0 + 6.93i)T + (144.5 + 250. i)T^{2}$$
19 $$1 + (13.7 - 7.91i)T + (180.5 - 312. i)T^{2}$$
23 $$1 + (18.2 + 31.6i)T + (-264.5 + 458. i)T^{2}$$
29 $$1 + 28.4T + 841T^{2}$$
31 $$1 + (36.2 + 20.9i)T + (480.5 + 832. i)T^{2}$$
37 $$1 + (7.14 + 12.3i)T + (-684.5 + 1.18e3i)T^{2}$$
41 $$1 - 21.3iT - 1.68e3T^{2}$$
43 $$1 - 55.3T + 1.84e3T^{2}$$
47 $$1 + (-29.3 + 16.9i)T + (1.10e3 - 1.91e3i)T^{2}$$
53 $$1 + (-42.4 + 73.4i)T + (-1.40e3 - 2.43e3i)T^{2}$$
59 $$1 + (-58.5 - 33.7i)T + (1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (25.6 - 14.7i)T + (1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (-27.4 + 47.5i)T + (-2.24e3 - 3.88e3i)T^{2}$$
71 $$1 - 83.8T + 5.04e3T^{2}$$
73 $$1 + (108. + 62.8i)T + (2.66e3 + 4.61e3i)T^{2}$$
79 $$1 + (35.1 + 60.9i)T + (-3.12e3 + 5.40e3i)T^{2}$$
83 $$1 - 27.1iT - 6.88e3T^{2}$$
89 $$1 + (126. - 73.0i)T + (3.96e3 - 6.85e3i)T^{2}$$
97 $$1 + 11.3iT - 9.40e3T^{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

−11.67179822385621102741769389406, −10.61552768147595045602374407417, −9.559927113595721830838655277331, −8.832862375478211798899016689916, −7.19728352329733238064851505685, −6.29579451147091094893429157359, −5.56796417033152320554905843528, −4.01522689123903629744745386687, −2.15218306056580456671119052096, −0.20429724425088123869977150741, 2.39464673530070294548699442503, 3.92766765019861120109434256628, 5.54771097453711238782354005211, 6.06562362104850041397893943086, 7.28910299154727172405806097537, 8.776642621522648082234784058281, 9.824495783817766097450362390895, 10.57089444487336427380276753774, 11.28110242650569964689458098434, 12.54938011365673858012354975323