# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 59$ Sign $0.867 - 0.497i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − 2-s + (−1.43 + 0.975i)3-s + 4-s − 0.0999i·5-s + (1.43 − 0.975i)6-s + 2.48·7-s − 8-s + (1.09 − 2.79i)9-s + 0.0999i·10-s − 1.50·11-s + (−1.43 + 0.975i)12-s − 3.74i·13-s − 2.48·14-s + (0.0974 + 0.143i)15-s + 16-s + 6.68i·17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (−0.826 + 0.563i)3-s + 0.5·4-s − 0.0446i·5-s + (0.584 − 0.398i)6-s + 0.939·7-s − 0.353·8-s + (0.365 − 0.930i)9-s + 0.0315i·10-s − 0.453·11-s + (−0.413 + 0.281i)12-s − 1.03i·13-s − 0.664·14-s + (0.0251 + 0.0369i)15-s + 0.250·16-s + 1.62i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$354$$    =    $$2 \cdot 3 \cdot 59$$ $$\varepsilon$$ = $0.867 - 0.497i$ motivic weight = $$1$$ character : $\chi_{354} (353, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 354,\ (\ :1/2),\ 0.867 - 0.497i)$ $L(1)$ $\approx$ $0.800002 + 0.212881i$ $L(\frac12)$ $\approx$ $0.800002 + 0.212881i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;59\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + (1.43 - 0.975i)T$$
59 $$1 + (-7.65 - 0.598i)T$$
good5 $$1 + 0.0999iT - 5T^{2}$$
7 $$1 - 2.48T + 7T^{2}$$
11 $$1 + 1.50T + 11T^{2}$$
13 $$1 + 3.74iT - 13T^{2}$$
17 $$1 - 6.68iT - 17T^{2}$$
19 $$1 - 3.99T + 19T^{2}$$
23 $$1 - 7.31T + 23T^{2}$$
29 $$1 + 3.18iT - 29T^{2}$$
31 $$1 - 9.23iT - 31T^{2}$$
37 $$1 + 0.665iT - 37T^{2}$$
41 $$1 + 3.99iT - 41T^{2}$$
43 $$1 - 10.0iT - 43T^{2}$$
47 $$1 - 4.30T + 47T^{2}$$
53 $$1 + 0.250iT - 53T^{2}$$
61 $$1 + 13.2iT - 61T^{2}$$
67 $$1 + 7.06iT - 67T^{2}$$
71 $$1 + 10.1iT - 71T^{2}$$
73 $$1 - 11.3iT - 73T^{2}$$
79 $$1 + 9.13T + 79T^{2}$$
83 $$1 - 5.43T + 83T^{2}$$
89 $$1 + 8.23T + 89T^{2}$$
97 $$1 - 3.83iT - 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

−11.04880464750286738869841662514, −10.80282297292399115717228291231, −9.893224881482748017396226711410, −8.758178959144775932034239080901, −7.939178463539502365175350072294, −6.80981651484537241887001326419, −5.59807638040890677414896667440, −4.82565996842722380661565077074, −3.23681724461727463686502788979, −1.18898358190915617803119260905, 1.05572735147348133659319009690, 2.53914572379603289372895817358, 4.71684794415080759783679346140, 5.52840726495451691589822231831, 7.04000358382428155309849432189, 7.32849414381893626722067905846, 8.578342082048149276589812942227, 9.536876475017593000485888984861, 10.67772622660984370747177048299, 11.48107862452705285700905108642