# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.191 - 0.981i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (−3.98 − 0.385i)2-s + (15.7 + 3.06i)4-s + 22.8·5-s + 82.1i·7-s + (−61.3 − 18.2i)8-s + (−90.8 − 8.79i)10-s − 107. i·11-s + 57.1·13-s + (31.6 − 326. i)14-s + (237. + 96.3i)16-s − 291.·17-s + 304. i·19-s + (358. + 69.9i)20-s + (−41.3 + 427. i)22-s + 982. i·23-s + ⋯
 L(s)  = 1 + (−0.995 − 0.0963i)2-s + (0.981 + 0.191i)4-s + 0.912·5-s + 1.67i·7-s + (−0.958 − 0.285i)8-s + (−0.908 − 0.0879i)10-s − 0.886i·11-s + 0.338·13-s + (0.161 − 1.66i)14-s + (0.926 + 0.376i)16-s − 1.01·17-s + 0.844i·19-s + (0.895 + 0.174i)20-s + (−0.0854 + 0.882i)22-s + 1.85i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.191 - 0.981i$ motivic weight = $$4$$ character : $\chi_{108} (55, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :2),\ 0.191 - 0.981i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.874842 + 0.720425i$$ $$L(\frac12)$$ $$\approx$$ $$0.874842 + 0.720425i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (3.98 + 0.385i)T$$
3 $$1$$
good5 $$1 - 22.8T + 625T^{2}$$
7 $$1 - 82.1iT - 2.40e3T^{2}$$
11 $$1 + 107. iT - 1.46e4T^{2}$$
13 $$1 - 57.1T + 2.85e4T^{2}$$
17 $$1 + 291.T + 8.35e4T^{2}$$
19 $$1 - 304. iT - 1.30e5T^{2}$$
23 $$1 - 982. iT - 2.79e5T^{2}$$
29 $$1 - 1.04e3T + 7.07e5T^{2}$$
31 $$1 - 783. iT - 9.23e5T^{2}$$
37 $$1 - 1.40e3T + 1.87e6T^{2}$$
41 $$1 - 363.T + 2.82e6T^{2}$$
43 $$1 + 64.7iT - 3.41e6T^{2}$$
47 $$1 - 3.49e3iT - 4.87e6T^{2}$$
53 $$1 + 3.26e3T + 7.89e6T^{2}$$
59 $$1 + 867. iT - 1.21e7T^{2}$$
61 $$1 - 6.03e3T + 1.38e7T^{2}$$
67 $$1 + 3.42e3iT - 2.01e7T^{2}$$
71 $$1 - 3.18e3iT - 2.54e7T^{2}$$
73 $$1 + 4.26e3T + 2.83e7T^{2}$$
79 $$1 + 7.53e3iT - 3.89e7T^{2}$$
83 $$1 + 586. iT - 4.74e7T^{2}$$
89 $$1 + 5.78e3T + 6.27e7T^{2}$$
97 $$1 - 1.08e4T + 8.85e7T^{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

−13.05584911163756814341470360621, −11.90994054123274705361118489607, −11.06888240703528882631379105325, −9.733200230139926335400285320967, −9.004156991249712124549064408790, −8.107727716607807041666117171267, −6.31111182155729322409317239105, −5.65182151571346463244814808895, −2.93915408115188284564109645471, −1.67074873845682746378983813636, 0.69920590090473968277336336871, 2.29155572848525603302774112394, 4.46479231212125653614048863288, 6.40847122551659817951401062272, 7.12044226990776715576550823932, 8.451619223390909265151881874818, 9.731879636699035049628626455315, 10.36929571220774334450235403477, 11.27003108400850625818922696379, 12.85378058532308950447665502357