# Properties

 Degree 2 Conductor $2^{2}$ Sign $0.104 - 0.994i$ Motivic weight 36 Primitive yes Self-dual no Analytic rank 0

# Learn more about

## Dirichlet series

 L(s)  = 1 + (1.94e5 − 1.75e5i)2-s − 4.34e8i·3-s + (7.18e9 − 6.83e10i)4-s − 7.52e12·5-s + (−7.62e13 − 8.47e13i)6-s − 2.26e15i·7-s + (−1.05e16 − 1.45e16i)8-s − 3.89e16·9-s + (−1.46e18 + 1.31e18i)10-s − 3.43e17i·11-s + (−2.97e19 − 3.12e18i)12-s + 4.11e19·13-s + (−3.97e20 − 4.41e20i)14-s + 3.27e21i·15-s + (−4.61e21 − 9.82e20i)16-s + 1.36e22·17-s + ⋯
 L(s)  = 1 + (0.743 − 0.669i)2-s − 1.12i·3-s + (0.104 − 0.994i)4-s − 1.97·5-s + (−0.751 − 0.834i)6-s − 1.39i·7-s + (−0.587 − 0.809i)8-s − 0.259·9-s + (−1.46 + 1.31i)10-s − 0.0617i·11-s + (−1.11 − 0.117i)12-s + 0.365·13-s + (−0.930 − 1.03i)14-s + 2.21i·15-s + (−0.978 − 0.208i)16-s + 0.972·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(37-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4$$    =    $$2^{2}$$ $$\varepsilon$$ = $0.104 - 0.994i$ motivic weight = $$36$$ character : $\chi_{4} (3, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 4,\ (\ :18),\ 0.104 - 0.994i)$$ $$L(\frac{37}{2})$$ $$\approx$$ $$1.321501159$$ $$L(\frac12)$$ $$\approx$$ $$1.321501159$$ $$L(19)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-1.94e5 + 1.75e5i)T$$
good3 $$1 + 4.34e8iT - 1.50e17T^{2}$$
5 $$1 + 7.52e12T + 1.45e25T^{2}$$
7 $$1 + 2.26e15iT - 2.65e30T^{2}$$
11 $$1 + 3.43e17iT - 3.09e37T^{2}$$
13 $$1 - 4.11e19T + 1.26e40T^{2}$$
17 $$1 - 1.36e22T + 1.97e44T^{2}$$
19 $$1 - 6.68e21iT - 1.08e46T^{2}$$
23 $$1 + 3.00e24iT - 1.05e49T^{2}$$
29 $$1 + 1.64e26T + 4.42e52T^{2}$$
31 $$1 - 4.32e26iT - 4.88e53T^{2}$$
37 $$1 + 2.47e27T + 2.85e56T^{2}$$
41 $$1 - 3.98e28T + 1.14e58T^{2}$$
43 $$1 - 3.42e29iT - 6.38e58T^{2}$$
47 $$1 + 7.98e29iT - 1.56e60T^{2}$$
53 $$1 + 1.60e30T + 1.18e62T^{2}$$
59 $$1 + 8.75e31iT - 5.63e63T^{2}$$
61 $$1 + 3.85e31T + 1.87e64T^{2}$$
67 $$1 - 7.88e32iT - 5.47e65T^{2}$$
71 $$1 + 3.14e33iT - 4.41e66T^{2}$$
73 $$1 + 3.43e33T + 1.20e67T^{2}$$
79 $$1 - 9.73e33iT - 2.06e68T^{2}$$
83 $$1 + 3.05e34iT - 1.22e69T^{2}$$
89 $$1 - 1.54e35T + 1.50e70T^{2}$$
97 $$1 + 2.22e35T + 3.34e71T^{2}$$
show more
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

−14.50667635697009270839213499528, −13.00066681753523727268404649836, −11.94368392108770701218222500317, −10.71870533262132719444865140092, −7.85730878484204942387844550259, −6.83837428417473833291642958683, −4.37859950966259184475433839311, −3.35417896003519484497123416613, −1.18036922855491536329562800152, −0.36659457287038227066141878196, 3.20729343278486458351547952298, 4.12574245572111724383286374452, 5.42061816630855606782404890278, 7.57338638196792409140670048807, 8.876792391505694212027050236118, 11.36706225754926965615087341097, 12.37292640099375414632340430704, 14.96009602644291115044848295645, 15.50382533338031385943304480770, 16.36781275879018641886417701543