Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.774 - 0.632i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (1.84 + 0.778i)2-s + (2.78 + 2.86i)4-s + (−1.10 − 1.90i)5-s + (7.23 + 4.17i)7-s + (2.90 + 7.45i)8-s + (−0.544 − 4.36i)10-s + (4.54 + 2.62i)11-s + (−7.37 − 12.7i)13-s + (10.0 + 13.3i)14-s + (−0.450 + 15.9i)16-s − 28.2·17-s − 19.1i·19-s + (2.39 − 8.47i)20-s + (6.33 + 8.37i)22-s + (−3.16 + 1.82i)23-s + ⋯
 L(s)  = 1 + (0.921 + 0.389i)2-s + (0.697 + 0.716i)4-s + (−0.220 − 0.381i)5-s + (1.03 + 0.597i)7-s + (0.363 + 0.931i)8-s + (−0.0544 − 0.436i)10-s + (0.413 + 0.238i)11-s + (−0.567 − 0.982i)13-s + (0.720 + 0.952i)14-s + (−0.0281 + 0.999i)16-s − 1.66·17-s − 1.00i·19-s + (0.119 − 0.423i)20-s + (0.287 + 0.380i)22-s + (−0.137 + 0.0794i)23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.774 - 0.632i$ motivic weight = $$2$$ character : $\chi_{108} (19, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :1),\ 0.774 - 0.632i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$2.14986 + 0.765714i$$ $$L(\frac12)$$ $$\approx$$ $$2.14986 + 0.765714i$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ 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 + (-1.84 - 0.778i)T$$
3 $$1$$
good5 $$1 + (1.10 + 1.90i)T + (-12.5 + 21.6i)T^{2}$$
7 $$1 + (-7.23 - 4.17i)T + (24.5 + 42.4i)T^{2}$$
11 $$1 + (-4.54 - 2.62i)T + (60.5 + 104. i)T^{2}$$
13 $$1 + (7.37 + 12.7i)T + (-84.5 + 146. i)T^{2}$$
17 $$1 + 28.2T + 289T^{2}$$
19 $$1 + 19.1iT - 361T^{2}$$
23 $$1 + (3.16 - 1.82i)T + (264.5 - 458. i)T^{2}$$
29 $$1 + (-12.3 + 21.3i)T + (-420.5 - 728. i)T^{2}$$
31 $$1 + (32.9 - 19.0i)T + (480.5 - 832. i)T^{2}$$
37 $$1 + 4.21T + 1.36e3T^{2}$$
41 $$1 + (-9.92 - 17.1i)T + (-840.5 + 1.45e3i)T^{2}$$
43 $$1 + (-20.1 - 11.6i)T + (924.5 + 1.60e3i)T^{2}$$
47 $$1 + (25.8 + 14.9i)T + (1.10e3 + 1.91e3i)T^{2}$$
53 $$1 - 32.1T + 2.80e3T^{2}$$
59 $$1 + (-7.96 + 4.59i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (40.8 - 70.7i)T + (-1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (-6.86 + 3.96i)T + (2.24e3 - 3.88e3i)T^{2}$$
71 $$1 - 62.9iT - 5.04e3T^{2}$$
73 $$1 - 33.3T + 5.32e3T^{2}$$
79 $$1 + (-53.7 - 31.0i)T + (3.12e3 + 5.40e3i)T^{2}$$
83 $$1 + (-103. - 59.4i)T + (3.44e3 + 5.96e3i)T^{2}$$
89 $$1 - 107.T + 7.92e3T^{2}$$
97 $$1 + (-1.78 + 3.09i)T + (-4.70e3 - 8.14e3i)T^{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

−13.56531073068516567490147160180, −12.60671573938233310630384422344, −11.72343128070076313602980222235, −10.82547171837280879280131518741, −8.923680315405402432158330954324, −7.986600792637740619277805022258, −6.72610922315317839455316533705, −5.25485989631086839116026862832, −4.40027408612530290173247774511, −2.41955482269049163010347208014, 1.90088102603402957475671973919, 3.84709874097132258207358421828, 4.87176358146779922439016780308, 6.49566095960801606467864287849, 7.50615865741608720606512709434, 9.177312134236502288321837576840, 10.70380400475800851327720926592, 11.25238010116248575423535023202, 12.23701520284924939760855732897, 13.49786089941613009940647349474