# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-0.933 + 0.358i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (3.05 − 2.57i)2-s + (2.72 − 15.7i)4-s + (−14.3 − 24.7i)5-s + (22.2 + 12.8i)7-s + (−32.3 − 55.2i)8-s + (−107. − 38.9i)10-s + (−93.9 − 54.2i)11-s + (44.2 + 76.5i)13-s + (101. − 17.9i)14-s + (−241. − 85.8i)16-s − 504.·17-s − 191. i·19-s + (−429. + 158. i)20-s + (−427. + 76.1i)22-s + (831. − 480. i)23-s + ⋯
 L(s)  = 1 + (0.764 − 0.644i)2-s + (0.170 − 0.985i)4-s + (−0.572 − 0.991i)5-s + (0.453 + 0.261i)7-s + (−0.504 − 0.863i)8-s + (−1.07 − 0.389i)10-s + (−0.776 − 0.448i)11-s + (0.261 + 0.453i)13-s + (0.515 − 0.0918i)14-s + (−0.942 − 0.335i)16-s − 1.74·17-s − 0.530i·19-s + (−1.07 + 0.395i)20-s + (−0.882 + 0.157i)22-s + (1.57 − 0.907i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.933 + 0.358i$ motivic weight = $$4$$ character : $\chi_{108} (19, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :2),\ -0.933 + 0.358i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.348227 - 1.87938i$$ $$L(\frac12)$$ $$\approx$$ $$0.348227 - 1.87938i$$ $$L(3)$$ 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 + (-3.05 + 2.57i)T$$
3 $$1$$
good5 $$1 + (14.3 + 24.7i)T + (-312.5 + 541. i)T^{2}$$
7 $$1 + (-22.2 - 12.8i)T + (1.20e3 + 2.07e3i)T^{2}$$
11 $$1 + (93.9 + 54.2i)T + (7.32e3 + 1.26e4i)T^{2}$$
13 $$1 + (-44.2 - 76.5i)T + (-1.42e4 + 2.47e4i)T^{2}$$
17 $$1 + 504.T + 8.35e4T^{2}$$
19 $$1 + 191. iT - 1.30e5T^{2}$$
23 $$1 + (-831. + 480. i)T + (1.39e5 - 2.42e5i)T^{2}$$
29 $$1 + (-396. + 687. i)T + (-3.53e5 - 6.12e5i)T^{2}$$
31 $$1 + (285. - 164. i)T + (4.61e5 - 7.99e5i)T^{2}$$
37 $$1 - 209.T + 1.87e6T^{2}$$
41 $$1 + (-528. - 914. i)T + (-1.41e6 + 2.44e6i)T^{2}$$
43 $$1 + (-2.88e3 - 1.66e3i)T + (1.70e6 + 2.96e6i)T^{2}$$
47 $$1 + (-977. - 564. i)T + (2.43e6 + 4.22e6i)T^{2}$$
53 $$1 + 1.13e3T + 7.89e6T^{2}$$
59 $$1 + (-4.03e3 + 2.33e3i)T + (6.05e6 - 1.04e7i)T^{2}$$
61 $$1 + (-2.79e3 + 4.84e3i)T + (-6.92e6 - 1.19e7i)T^{2}$$
67 $$1 + (-6.12e3 + 3.53e3i)T + (1.00e7 - 1.74e7i)T^{2}$$
71 $$1 - 4.43e3iT - 2.54e7T^{2}$$
73 $$1 + 1.95e3T + 2.83e7T^{2}$$
79 $$1 + (1.52e3 + 879. i)T + (1.94e7 + 3.37e7i)T^{2}$$
83 $$1 + (2.62e3 + 1.51e3i)T + (2.37e7 + 4.11e7i)T^{2}$$
89 $$1 - 559.T + 6.27e7T^{2}$$
97 $$1 + (-1.10e3 + 1.90e3i)T + (-4.42e7 - 7.66e7i)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

−12.72574874684154893569735902433, −11.42346481220820113230188175165, −10.92281455808950614095220855338, −9.243806598389651254007357554890, −8.363325964487692113425673874478, −6.60788093935198423327043695747, −5.06868609070256536020978165305, −4.32610788062708071370311770569, −2.49279553383707931645145450583, −0.65859066781221928549747609964, 2.68809270268556389292393620842, 4.04895547609738533655711272473, 5.37065342996470640757533284895, 6.90697896588268558947198241150, 7.52883291767212073974997069600, 8.789025963538064972841674084327, 10.72839806190527451696355288332, 11.28853940207651931277128480272, 12.66416167122806011578200541622, 13.52422124295123544839435121662