# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 67$ Sign $-0.992 + 0.125i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (1.01 + 0.985i)2-s − 3-s + (0.0573 + 1.99i)4-s − 1.66i·5-s + (−1.01 − 0.985i)6-s − 0.193·7-s + (−1.91 + 2.08i)8-s + 9-s + (1.64 − 1.69i)10-s − 5.43·11-s + (−0.0573 − 1.99i)12-s + 3.51i·13-s + (−0.196 − 0.190i)14-s + 1.66i·15-s + (−3.99 + 0.229i)16-s − 4.63·17-s + ⋯
 L(s)  = 1 + (0.717 + 0.696i)2-s − 0.577·3-s + (0.0286 + 0.999i)4-s − 0.746i·5-s + (−0.414 − 0.402i)6-s − 0.0731·7-s + (−0.676 + 0.736i)8-s + 0.333·9-s + (0.520 − 0.535i)10-s − 1.63·11-s + (−0.0165 − 0.577i)12-s + 0.975i·13-s + (−0.0524 − 0.0510i)14-s + 0.430i·15-s + (−0.998 + 0.0573i)16-s − 1.12·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$804$$    =    $$2^{2} \cdot 3 \cdot 67$$ $$\varepsilon$$ = $-0.992 + 0.125i$ motivic weight = $$1$$ character : $\chi_{804} (535, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 804,\ (\ :1/2),\ -0.992 + 0.125i)$ $L(1)$ $\approx$ $0.0438479 - 0.698500i$ $L(\frac12)$ $\approx$ $0.0438479 - 0.698500i$ $L(\frac{3}{2})$ 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 + (-1.01 - 0.985i)T$$
3 $$1 + T$$
67 $$1 + (0.790 + 8.14i)T$$
good5 $$1 + 1.66iT - 5T^{2}$$
7 $$1 + 0.193T + 7T^{2}$$
11 $$1 + 5.43T + 11T^{2}$$
13 $$1 - 3.51iT - 13T^{2}$$
17 $$1 + 4.63T + 17T^{2}$$
19 $$1 - 1.14iT - 19T^{2}$$
23 $$1 - 7.82iT - 23T^{2}$$
29 $$1 + 3.87T + 29T^{2}$$
31 $$1 + 8.23T + 31T^{2}$$
37 $$1 - 11.0T + 37T^{2}$$
41 $$1 - 0.746iT - 41T^{2}$$
43 $$1 + 3.05T + 43T^{2}$$
47 $$1 + 2.28iT - 47T^{2}$$
53 $$1 + 6.58iT - 53T^{2}$$
59 $$1 - 1.92iT - 59T^{2}$$
61 $$1 + 1.81iT - 61T^{2}$$
71 $$1 - 15.3iT - 71T^{2}$$
73 $$1 - 5.05T + 73T^{2}$$
79 $$1 - 8.43T + 79T^{2}$$
83 $$1 + 8.12iT - 83T^{2}$$
89 $$1 - 4.29T + 89T^{2}$$
97 $$1 - 8.73iT - 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

−11.08528389871207457114806980311, −9.691470577048294126639475127815, −8.896743075108490047588836856072, −7.900630046556476629314575117402, −7.20441297415273002413061846845, −6.17682531675347027523421349921, −5.28481463207772592793115684477, −4.74703294400633446648541915703, −3.64713851652245961323138826768, −2.12121593870486800544132675360, 0.26991393292343422336989653963, 2.31883425424630559862041107346, 3.06152112011141986985370379020, 4.42774642304216480413988130310, 5.25472590526171154512820218876, 6.10744267930457843856575533486, 6.96005478201978937434592829044, 8.000646885969651410015421346846, 9.290961858255623400476794616957, 10.30679415821408558082472759226