# Properties

 Degree 2 Conductor $3 \cdot 19$ Sign $0.583 + 0.812i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.49 − 1.25i)2-s + (−1.72 − 0.130i)3-s + (0.317 − 1.79i)4-s + (0.487 − 0.0858i)5-s + (−2.75 + 1.97i)6-s + (−0.969 + 1.67i)7-s + (0.170 + 0.295i)8-s + (2.96 + 0.449i)9-s + (0.621 − 0.741i)10-s + (−3.99 + 2.30i)11-s + (−0.781 + 3.06i)12-s + (−1.79 − 4.92i)13-s + (0.658 + 3.73i)14-s + (−0.852 + 0.0849i)15-s + (4.05 + 1.47i)16-s + (−1.61 − 1.92i)17-s + ⋯
 L(s)  = 1 + (1.05 − 0.889i)2-s + (−0.997 − 0.0751i)3-s + (0.158 − 0.898i)4-s + (0.217 − 0.0384i)5-s + (−1.12 + 0.806i)6-s + (−0.366 + 0.634i)7-s + (0.0602 + 0.104i)8-s + (0.988 + 0.149i)9-s + (0.196 − 0.234i)10-s + (−1.20 + 0.695i)11-s + (−0.225 + 0.884i)12-s + (−0.497 − 1.36i)13-s + (0.176 + 0.998i)14-s + (−0.220 + 0.0219i)15-s + (1.01 + 0.369i)16-s + (−0.392 − 0.467i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$57$$    =    $$3 \cdot 19$$ $$\varepsilon$$ = $0.583 + 0.812i$ motivic weight = $$1$$ character : $\chi_{57} (32, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 57,\ (\ :1/2),\ 0.583 + 0.812i)$ $L(1)$ $\approx$ $0.950206 - 0.487687i$ $L(\frac12)$ $\approx$ $0.950206 - 0.487687i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;19\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 + (1.72 + 0.130i)T$$
19 $$1 + (-2.80 + 3.33i)T$$
good2 $$1 + (-1.49 + 1.25i)T + (0.347 - 1.96i)T^{2}$$
5 $$1 + (-0.487 + 0.0858i)T + (4.69 - 1.71i)T^{2}$$
7 $$1 + (0.969 - 1.67i)T + (-3.5 - 6.06i)T^{2}$$
11 $$1 + (3.99 - 2.30i)T + (5.5 - 9.52i)T^{2}$$
13 $$1 + (1.79 + 4.92i)T + (-9.95 + 8.35i)T^{2}$$
17 $$1 + (1.61 + 1.92i)T + (-2.95 + 16.7i)T^{2}$$
23 $$1 + (-3.35 - 0.591i)T + (21.6 + 7.86i)T^{2}$$
29 $$1 + (-0.872 - 0.731i)T + (5.03 + 28.5i)T^{2}$$
31 $$1 + (1.31 + 0.758i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 - 3.28iT - 37T^{2}$$
41 $$1 + (-9.40 - 3.42i)T + (31.4 + 26.3i)T^{2}$$
43 $$1 + (0.829 + 4.70i)T + (-40.4 + 14.7i)T^{2}$$
47 $$1 + (3.82 - 4.56i)T + (-8.16 - 46.2i)T^{2}$$
53 $$1 + (-1.71 + 9.73i)T + (-49.8 - 18.1i)T^{2}$$
59 $$1 + (0.172 - 0.144i)T + (10.2 - 58.1i)T^{2}$$
61 $$1 + (1.90 - 10.7i)T + (-57.3 - 20.8i)T^{2}$$
67 $$1 + (-0.266 + 0.317i)T + (-11.6 - 65.9i)T^{2}$$
71 $$1 + (-0.540 - 3.06i)T + (-66.7 + 24.2i)T^{2}$$
73 $$1 + (0.118 + 0.0430i)T + (55.9 + 46.9i)T^{2}$$
79 $$1 + (-4.94 + 13.5i)T + (-60.5 - 50.7i)T^{2}$$
83 $$1 + (6.05 + 3.49i)T + (41.5 + 71.8i)T^{2}$$
89 $$1 + (5.32 - 1.93i)T + (68.1 - 57.2i)T^{2}$$
97 $$1 + (4.49 + 5.36i)T + (-16.8 + 95.5i)T^{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

−15.08306307627369839327510555668, −13.28586214026345824809607827605, −12.85204794717140138906475216332, −11.86988687432196270893073472489, −10.82029642037965464458414705621, −9.796686614240549101831128638647, −7.53531077763059156115958000768, −5.63971115786942479940481375468, −4.88085827272819601208854532071, −2.70849373730960705652318018339, 4.10118998631940605957998059636, 5.40028307746756552605898902890, 6.45276109467892519927219037987, 7.53753238418708037631996499533, 9.812291823582266727609811885541, 10.95050458250366087528218885024, 12.37169690418043484265383472659, 13.35119861966610712165320532726, 14.18866870526732604962959149347, 15.58225653132493461290092865839