# Properties

 Degree 2 Conductor $3^{2} \cdot 13$ Sign $0.957 - 0.289i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + i·4-s + (−1 − i)7-s − i·13-s − 16-s + (−1 + i)19-s + i·25-s + (1 − i)28-s + (1 − i)31-s + (1 + i)37-s + i·49-s + 52-s − i·64-s + (−1 + i)67-s + (−1 − i)73-s + (−1 − i)76-s + ⋯
 L(s)  = 1 + i·4-s + (−1 − i)7-s − i·13-s − 16-s + (−1 + i)19-s + i·25-s + (1 − i)28-s + (1 − i)31-s + (1 + i)37-s + i·49-s + 52-s − i·64-s + (−1 + i)67-s + (−1 − i)73-s + (−1 − i)76-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$117$$    =    $$3^{2} \cdot 13$$ $$\varepsilon$$ = $0.957 - 0.289i$ motivic weight = $$0$$ character : $\chi_{117} (109, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 117,\ (\ :0),\ 0.957 - 0.289i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.5569436756$$ $$L(\frac12)$$ $$\approx$$ $$0.5569436756$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;13\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1$$
13 $$1 + iT$$
good2 $$1 - iT^{2}$$
5 $$1 - iT^{2}$$
7 $$1 + (1 + i)T + iT^{2}$$
11 $$1 + iT^{2}$$
17 $$1 - T^{2}$$
19 $$1 + (1 - i)T - iT^{2}$$
23 $$1 - T^{2}$$
29 $$1 + T^{2}$$
31 $$1 + (-1 + i)T - iT^{2}$$
37 $$1 + (-1 - i)T + iT^{2}$$
41 $$1 - iT^{2}$$
43 $$1 - T^{2}$$
47 $$1 + iT^{2}$$
53 $$1 + T^{2}$$
59 $$1 + iT^{2}$$
61 $$1 + T^{2}$$
67 $$1 + (1 - i)T - iT^{2}$$
71 $$1 - iT^{2}$$
73 $$1 + (1 + i)T + iT^{2}$$
79 $$1 + T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + iT^{2}$$
97 $$1 + (-1 + i)T - iT^{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

−13.36001136281506947321561544962, −13.05270677275964737188948525381, −11.93499686341665259569750108108, −10.65262963256653539214358760704, −9.701049372459152198528131483308, −8.287424301189132676813928040431, −7.37991611206024721121675636931, −6.18811586118836933892295665132, −4.21585912692895745454493259751, −3.10156635655119368499674695018, 2.45453865405386386796231817165, 4.57945296969333583056588158114, 6.02436325627031227717949276801, 6.74227370788945589021286031122, 8.744582829751551591102352480621, 9.460599077616326652932631890719, 10.52450589542211795910627174439, 11.67036785348349858475122944719, 12.72164914596868163355269831842, 13.80990494819017442544207341531