# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 67$ Sign $-0.609 + 0.792i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.936 − 1.45i)3-s + (0.686 + 0.313i)7-s + (−1.24 + 2.72i)9-s + (1.32 − 4.50i)13-s + (−3.57 − 7.82i)19-s + (−0.185 − 1.29i)21-s + (4.79 + 1.40i)25-s + (5.14 − 0.739i)27-s + (−0.698 − 2.37i)31-s − 11.5·37-s + (−7.80 + 2.29i)39-s + (−9.23 − 8.00i)43-s + (−4.21 − 4.86i)49-s + (−8.05 + 12.5i)57-s + (13.2 − 1.90i)61-s + ⋯
 L(s)  = 1 + (−0.540 − 0.841i)3-s + (0.259 + 0.118i)7-s + (−0.415 + 0.909i)9-s + (0.367 − 1.25i)13-s + (−0.819 − 1.79i)19-s + (−0.0405 − 0.282i)21-s + (0.959 + 0.281i)25-s + (0.989 − 0.142i)27-s + (−0.125 − 0.427i)31-s − 1.89·37-s + (−1.25 + 0.367i)39-s + (−1.40 − 1.22i)43-s + (−0.601 − 0.694i)49-s + (−1.06 + 1.66i)57-s + (1.69 − 0.244i)61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$804$$    =    $$2^{2} \cdot 3 \cdot 67$$ $$\varepsilon$$ = $-0.609 + 0.792i$ motivic weight = $$1$$ character : $\chi_{804} (713, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 804,\ (\ :1/2),\ -0.609 + 0.792i)$ $L(1)$ $\approx$ $0.430625 - 0.874834i$ $L(\frac12)$ $\approx$ $0.430625 - 0.874834i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;67\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (0.936 + 1.45i)T$$
67 $$1 + (7.18 + 3.91i)T$$
good5 $$1 + (-4.79 - 1.40i)T^{2}$$
7 $$1 + (-0.686 - 0.313i)T + (4.58 + 5.29i)T^{2}$$
11 $$1 + (-10.5 - 3.09i)T^{2}$$
13 $$1 + (-1.32 + 4.50i)T + (-10.9 - 7.02i)T^{2}$$
17 $$1 + (2.41 + 16.8i)T^{2}$$
19 $$1 + (3.57 + 7.82i)T + (-12.4 + 14.3i)T^{2}$$
23 $$1 + (-9.55 + 20.9i)T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + (0.698 + 2.37i)T + (-26.0 + 16.7i)T^{2}$$
37 $$1 + 11.5T + 37T^{2}$$
41 $$1 + (-5.83 - 40.5i)T^{2}$$
43 $$1 + (9.23 + 8.00i)T + (6.11 + 42.5i)T^{2}$$
47 $$1 + (-19.5 + 42.7i)T^{2}$$
53 $$1 + (-7.54 + 52.4i)T^{2}$$
59 $$1 + (-49.6 + 31.8i)T^{2}$$
61 $$1 + (-13.2 + 1.90i)T + (58.5 - 17.1i)T^{2}$$
71 $$1 + (10.1 - 70.2i)T^{2}$$
73 $$1 + (-2.15 - 14.9i)T + (-70.0 + 20.5i)T^{2}$$
79 $$1 + (-4.99 + 17.0i)T + (-66.4 - 42.7i)T^{2}$$
83 $$1 + (79.6 + 23.3i)T^{2}$$
89 $$1 + (-36.9 - 80.9i)T^{2}$$
97 $$1 - 4.08iT - 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

−10.23805190802194476232765062306, −8.803718711215544336986250585609, −8.313980488391664220224142552191, −7.19200143941364873635099610861, −6.62023466263329346705610667476, −5.47997721503969984397763124713, −4.86399647997287368104665638917, −3.24163032098975185337678331793, −2.03983713517344529464595399512, −0.52449979567729259925374496248, 1.64031235169965337597291560820, 3.40206171787609144113767829355, 4.26255876158784994377756764460, 5.11790575367471966500193673441, 6.18833168909016534732373230577, 6.85678960969084207486037384313, 8.255302302096136298549601700207, 8.891666395601446122853560587484, 9.886088045166193116377739543204, 10.52004413115373393660138584523