# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 67$ Sign $0.798 - 0.602i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.40 + 0.159i)2-s + 3-s + (1.94 − 0.449i)4-s − 0.947i·5-s + (−1.40 + 0.159i)6-s − 0.0251·7-s + (−2.66 + 0.943i)8-s + 9-s + (0.151 + 1.33i)10-s − 1.77·11-s + (1.94 − 0.449i)12-s + 5.89i·13-s + (0.0352 − 0.00401i)14-s − 0.947i·15-s + (3.59 − 1.75i)16-s + 2.08·17-s + ⋯
 L(s)  = 1 + (−0.993 + 0.113i)2-s + 0.577·3-s + (0.974 − 0.224i)4-s − 0.423i·5-s + (−0.573 + 0.0652i)6-s − 0.00949·7-s + (−0.942 + 0.333i)8-s + 0.333·9-s + (0.0478 + 0.420i)10-s − 0.536·11-s + (0.562 − 0.129i)12-s + 1.63i·13-s + (0.00943 − 0.00107i)14-s − 0.244i·15-s + (0.899 − 0.437i)16-s + 0.505·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$804$$    =    $$2^{2} \cdot 3 \cdot 67$$ $$\varepsilon$$ = $0.798 - 0.602i$ motivic weight = $$1$$ character : $\chi_{804} (535, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 804,\ (\ :1/2),\ 0.798 - 0.602i)$ $L(1)$ $\approx$ $1.12857 + 0.378227i$ $L(\frac12)$ $\approx$ $1.12857 + 0.378227i$ $L(\frac{3}{2})$ 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 + (1.40 - 0.159i)T$$
3 $$1 - T$$
67 $$1 + (-7.47 + 3.33i)T$$
good5 $$1 + 0.947iT - 5T^{2}$$
7 $$1 + 0.0251T + 7T^{2}$$
11 $$1 + 1.77T + 11T^{2}$$
13 $$1 - 5.89iT - 13T^{2}$$
17 $$1 - 2.08T + 17T^{2}$$
19 $$1 - 4.96iT - 19T^{2}$$
23 $$1 - 4.05iT - 23T^{2}$$
29 $$1 - 5.86T + 29T^{2}$$
31 $$1 - 3.69T + 31T^{2}$$
37 $$1 - 5.82T + 37T^{2}$$
41 $$1 + 6.60iT - 41T^{2}$$
43 $$1 - 2.25T + 43T^{2}$$
47 $$1 + 11.4iT - 47T^{2}$$
53 $$1 - 7.44iT - 53T^{2}$$
59 $$1 + 2.53iT - 59T^{2}$$
61 $$1 + 2.42iT - 61T^{2}$$
71 $$1 - 1.82iT - 71T^{2}$$
73 $$1 + 9.08T + 73T^{2}$$
79 $$1 - 12.9T + 79T^{2}$$
83 $$1 - 14.6iT - 83T^{2}$$
89 $$1 + 9.99T + 89T^{2}$$
97 $$1 - 3.04iT - 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

−10.04234449617599863619075841960, −9.460884182651311527201201260168, −8.634446171600377293282008295853, −8.005741270192064601109159803503, −7.12099207625913073650103868070, −6.24688640246122683569351110139, −5.06310728234257693033095444550, −3.75415904160156714525309028061, −2.44875032074281081561618137515, −1.32308786165168914760434611037, 0.863539949730038268369293608585, 2.73842726617020628679025995683, 2.99845729539994135341945962623, 4.75827609942916303742057324716, 6.07421467723131204008762222768, 6.95991284984453724541003594115, 7.928308547386166816626362722579, 8.310579801898497684559346091778, 9.362863490967132124159948083407, 10.20158347073389471751066826490