# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-0.146 + 0.989i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.95 + 2.27i)3-s + (−3.25 − 8.93i)5-s + (0.410 + 2.32i)7-s + (−1.38 − 8.89i)9-s + (4.40 − 12.1i)11-s + (−12.2 − 10.2i)13-s + (26.6 + 10.0i)15-s + (12.2 + 7.04i)17-s + (−3.29 − 5.70i)19-s + (−6.09 − 3.60i)21-s + (−25.9 − 4.58i)23-s + (−50.0 + 41.9i)25-s + (22.9 + 14.2i)27-s + (0.977 + 1.16i)29-s + (−0.620 + 3.52i)31-s + ⋯
 L(s)  = 1 + (−0.650 + 0.759i)3-s + (−0.650 − 1.78i)5-s + (0.0585 + 0.332i)7-s + (−0.153 − 0.988i)9-s + (0.400 − 1.10i)11-s + (−0.939 − 0.788i)13-s + (1.77 + 0.668i)15-s + (0.717 + 0.414i)17-s + (−0.173 − 0.300i)19-s + (−0.290 − 0.171i)21-s + (−1.12 − 0.199i)23-s + (−2.00 + 1.67i)25-s + (0.850 + 0.526i)27-s + (0.0337 + 0.0401i)29-s + (−0.0200 + 0.113i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.146 + 0.989i$ motivic weight = $$2$$ character : $\chi_{108} (5, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :1),\ -0.146 + 0.989i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.471676 - 0.546727i$$ $$L(\frac12)$$ $$\approx$$ $$0.471676 - 0.546727i$$ $$L(2)$$ 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$$
3 $$1 + (1.95 - 2.27i)T$$
good5 $$1 + (3.25 + 8.93i)T + (-19.1 + 16.0i)T^{2}$$
7 $$1 + (-0.410 - 2.32i)T + (-46.0 + 16.7i)T^{2}$$
11 $$1 + (-4.40 + 12.1i)T + (-92.6 - 77.7i)T^{2}$$
13 $$1 + (12.2 + 10.2i)T + (29.3 + 166. i)T^{2}$$
17 $$1 + (-12.2 - 7.04i)T + (144.5 + 250. i)T^{2}$$
19 $$1 + (3.29 + 5.70i)T + (-180.5 + 312. i)T^{2}$$
23 $$1 + (25.9 + 4.58i)T + (497. + 180. i)T^{2}$$
29 $$1 + (-0.977 - 1.16i)T + (-146. + 828. i)T^{2}$$
31 $$1 + (0.620 - 3.52i)T + (-903. - 328. i)T^{2}$$
37 $$1 + (11.0 - 19.2i)T + (-684.5 - 1.18e3i)T^{2}$$
41 $$1 + (-31.4 + 37.5i)T + (-291. - 1.65e3i)T^{2}$$
43 $$1 + (-78.8 - 28.6i)T + (1.41e3 + 1.18e3i)T^{2}$$
47 $$1 + (34.9 - 6.16i)T + (2.07e3 - 755. i)T^{2}$$
53 $$1 + 65.8iT - 2.80e3T^{2}$$
59 $$1 + (17.2 + 47.3i)T + (-2.66e3 + 2.23e3i)T^{2}$$
61 $$1 + (11.5 + 65.6i)T + (-3.49e3 + 1.27e3i)T^{2}$$
67 $$1 + (-72.5 - 60.8i)T + (779. + 4.42e3i)T^{2}$$
71 $$1 + (-71.8 - 41.4i)T + (2.52e3 + 4.36e3i)T^{2}$$
73 $$1 + (47.8 + 82.9i)T + (-2.66e3 + 4.61e3i)T^{2}$$
79 $$1 + (25.9 - 21.8i)T + (1.08e3 - 6.14e3i)T^{2}$$
83 $$1 + (-7.14 - 8.50i)T + (-1.19e3 + 6.78e3i)T^{2}$$
89 $$1 + (8.83 - 5.10i)T + (3.96e3 - 6.85e3i)T^{2}$$
97 $$1 + (-66.7 - 24.2i)T + (7.20e3 + 6.04e3i)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

−12.73257259462957993613248469136, −12.19473724351329347223800071264, −11.25823849326988086730385116206, −9.879033282335512236476103373732, −8.868155538269986924645411485362, −7.982238435805699282845684107069, −5.86093257606209009729419720820, −5.00325928798118295211912756806, −3.80192954357316551459478186251, −0.57485318002860518000879171722, 2.34825679350537444399477677436, 4.21687112139819131911602392316, 6.13375582204157764238668958710, 7.26875177274501105806503418349, 7.54793464874844018365104938817, 9.829669313676575682914108060546, 10.73999068245297447158148574519, 11.76832526049485913580822554660, 12.32794095438381083727515703678, 14.06026757419974936841091164075