# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 67$ Sign $0.638 - 0.769i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.393 + 1.68i)3-s + (3.45 − 1.01i)5-s + (−0.403 − 0.349i)7-s + (−2.69 − 1.32i)9-s + (−0.843 + 0.247i)11-s + (2.49 + 3.88i)13-s + (0.352 + 6.22i)15-s + (6.49 + 0.934i)17-s + (−1.01 − 1.17i)19-s + (0.747 − 0.542i)21-s + (−1.87 + 0.854i)23-s + (6.68 − 4.29i)25-s + (3.29 − 4.01i)27-s + 3.54i·29-s + (2.94 − 4.58i)31-s + ⋯
 L(s)  = 1 + (−0.227 + 0.973i)3-s + (1.54 − 0.453i)5-s + (−0.152 − 0.132i)7-s + (−0.896 − 0.442i)9-s + (−0.254 + 0.0746i)11-s + (0.692 + 1.07i)13-s + (0.0909 + 1.60i)15-s + (1.57 + 0.226i)17-s + (−0.233 − 0.269i)19-s + (0.163 − 0.118i)21-s + (−0.390 + 0.178i)23-s + (1.33 − 0.859i)25-s + (0.634 − 0.773i)27-s + 0.658i·29-s + (0.529 − 0.824i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$804$$    =    $$2^{2} \cdot 3 \cdot 67$$ $$\varepsilon$$ = $0.638 - 0.769i$ motivic weight = $$1$$ character : $\chi_{804} (5, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 804,\ (\ :1/2),\ 0.638 - 0.769i)$ $L(1)$ $\approx$ $1.68547 + 0.791674i$ $L(\frac12)$ $\approx$ $1.68547 + 0.791674i$ $L(\frac{3}{2})$ 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 + (0.393 - 1.68i)T$$
67 $$1 + (-4.80 - 6.62i)T$$
good5 $$1 + (-3.45 + 1.01i)T + (4.20 - 2.70i)T^{2}$$
7 $$1 + (0.403 + 0.349i)T + (0.996 + 6.92i)T^{2}$$
11 $$1 + (0.843 - 0.247i)T + (9.25 - 5.94i)T^{2}$$
13 $$1 + (-2.49 - 3.88i)T + (-5.40 + 11.8i)T^{2}$$
17 $$1 + (-6.49 - 0.934i)T + (16.3 + 4.78i)T^{2}$$
19 $$1 + (1.01 + 1.17i)T + (-2.70 + 18.8i)T^{2}$$
23 $$1 + (1.87 - 0.854i)T + (15.0 - 17.3i)T^{2}$$
29 $$1 - 3.54iT - 29T^{2}$$
31 $$1 + (-2.94 + 4.58i)T + (-12.8 - 28.1i)T^{2}$$
37 $$1 - 6.30T + 37T^{2}$$
41 $$1 + (1.58 - 11.0i)T + (-39.3 - 11.5i)T^{2}$$
43 $$1 + (-4.21 - 0.605i)T + (41.2 + 12.1i)T^{2}$$
47 $$1 + (8.24 - 3.76i)T + (30.7 - 35.5i)T^{2}$$
53 $$1 + (0.880 + 6.12i)T + (-50.8 + 14.9i)T^{2}$$
59 $$1 + (-0.431 + 0.671i)T + (-24.5 - 53.6i)T^{2}$$
61 $$1 + (1.65 - 5.64i)T + (-51.3 - 32.9i)T^{2}$$
71 $$1 + (3.24 - 0.465i)T + (68.1 - 20.0i)T^{2}$$
73 $$1 + (4.50 + 1.32i)T + (61.4 + 39.4i)T^{2}$$
79 $$1 + (1.15 + 1.79i)T + (-32.8 + 71.8i)T^{2}$$
83 $$1 + (0.393 + 1.33i)T + (-69.8 + 44.8i)T^{2}$$
89 $$1 + (-0.163 - 0.0744i)T + (58.2 + 67.2i)T^{2}$$
97 $$1 + 3.07iT - 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

−9.953546477823164128174767499660, −9.828022467399469765844615262907, −8.971924109455692968926605366590, −8.071548937968350391153546030101, −6.50626686176829390603393866010, −5.90216176110038163202236357012, −5.10218844796771805174565431898, −4.13132366773891239860693456677, −2.87204846668248972004515545436, −1.43347185545216692535774152563, 1.13565319894448277783931474372, 2.35334420248202265982389163452, 3.25082742751120206157284732557, 5.27138697629791125602214685970, 5.87819288242519428644117882725, 6.41458545636088356764087085098, 7.56202879862628471973119532184, 8.285175278993612161254704898944, 9.388458221747083050912334913841, 10.27938788899755998745116278158